text: RELATIVE FUNCTORIALITY AND FUNCTIONAL EQUATIONS VIA TRACE FORMULAS 31 ∗ characterized by the fact that ϕ and e∅(ϕ) are “equal close to inﬁnity”, for ∈ S every ϕ (X). For a rigorous deﬁnition of what this means, and the con- § struction of the map, cf. [SV, 5]. Notice that there the asymptotics map was deﬁned for functions, but here we consider it as a morphism of measures; there is a canonical way by which a G-invariant measure on X induces a § G-invariant measure on X∅, s. [SV, 4.2], so there is no problem passing from functions to measures. S S The space +(X∅) was introduced in [DHS]: it is the closure of (X∅) S → M under a set of “scattering morphisms” S : (X∅) (X∅) parametrized w by the little Weyl group W of X. The scattering morphisms give rise to an S X action of W on +(X∅) by G-automorphisms, which semi-commute with X ∈ the action of A , i.e., for t A we have X X · − · t S f = S (w 1t f). w w (Recall that the action of A on measures has been normalized in (5.3).) ∗ X The image of e∅ is precisely the subspace of W -invariants, with W acting X X through the scattering operators — this the Paley–Wiener theorem [DHS, Theorem 1.8]. At this point I stress that the boundary degeneration X∅ should not be considered just as an abstract G-variety, but as a G-variety equipped with extra structure, which is obtained from X. This extra structure is encoded S in the scattering morphisms. The space +(X∅) is not determined by the abstract variety X∅, but by the variety plus the scattering morphisms, which depend on X. Now consider the morphisms induced by asymptotics: ∗ ⊗ ∗ S ⊗ S ∨ → S ⊗ S ∨ e∅ e∅ : (X) (X ) +(X∅)WX +(X∅ )WX. One can show that they descend to morphisms on the spaces of push- forward measures: E∗ S × ∨ → S × ∨ ∅ : (X X /G) +(X∅ X∅ /G). (5.5) S × ∨ Note that this is not automatic, because +(X∅ X∅ /G) cannot be iden- S × tiﬁed with the Gdiag-coinvariants of +(X∅ X∅). For example, when (cid:39) ∨ \ (cid:39) (cid:114) { } × ∨ (cid:39) \ X∅ X∅ = N SL A2 0 , so X∅ X∅ /G N G/N, the map (cid:114) { } → (cid:39) 2 A2 0 C A1 is the projection to a coordinate, but every evaluation ∈ on a point in the preimage of 0 C is an N-invariant distribution. Nonethe- ∗ less, using the invariance of the image of the asymptotics morphism e∅ un- E∗ der the scattering operators, one can show that the map ∅ descending from ∗ ⊗ ∗ e∅ e∅ does exist, uniquely, and one can compute the pull-backs of relative characters under this map. We will only present the ﬁnal result for the ex- amples considered in this paper. ∨ ∨ × Note that the boundary degeneration X∅ for X is isomorphic to A P(X) X × − G, whereas the boundary degeneration for X is A P(X) G. Thus, the X

text: RELATIVE

text: FUNCTORIALITY

text: AND

text: FUNCTIONAL

text: EQUATIONS

text: VIA

text: TRACE

text: FORMULAS

text: characterized

text: by

text: the

text: fact

text: that

text: ϕ

text: and

text: e∅(ϕ)

text: are

text: “equal

text: close

text: to

text: inﬁnity”,

text: for

text: S

text: every

text: ϕ

text: (X).

text: For

text: a

text: rigorous

text: deﬁnition

text: of

text: what

text: this

text: means,

text: and

text: the

text: con-

text: struction

text: of

text: the

text: map,

text: cf.

text: [SV,

text: Notice

text: that

text: there

text: the

text: asymptotics

text: map

text: was

text: deﬁned

text: for

text: functions,

text: but

text: here

text: we

text: consider

text: it

text: as

text: a

text: morphism

text: of

text: measures;

text: there

text: is

text: a

text: canonical

text: way

text: by

text: which

text: a

text: G-invariant

text: measure

text: on

text: X

text: induces

text: a

text: G-invariant

text: measure

text: on

text: X∅,

text: s.

text: [SV,

text: so

text: there

text: is

text: no

text: problem

text: passing

text: from

text: functions

text: to

text: measures.

text: S

text: S

text: The

text: space

text: +(X∅)

text: was

text: introduced

text: in

text: [DHS]:

text: it

text: is

text: the

text: closure

text: of

text: (X∅)

text: S

text: M

text: under

text: a

text: set

text: of

text: “scattering

text: morphisms”

text: S

text: (X∅)

text: (X∅)

text: parametrized

text: w

text: by

text: the

text: little

text: Weyl

text: group

text: W

text: of

text: X.

text: The

text: scattering

text: morphisms

text: give

text: rise

text: to

text: an

text: S

text: X

text: action

text: of

text: W

text: on

text: +(X∅)

text: by

text: G-automorphisms,

text: which

text: semi-commute

text: with

text: X

text: the

text: action

text: of

text: A

text: i.e.,

text: for

text: t

text: A

text: we

text: have

text: X

text: X

text: t

text: S

text: f

text: S

text: (w

text: f).

text: w

text: w

text: (Recall

text: that

text: the

text: action

text: of

text: A

text: on

text: measures

text: has

text: been

text: normalized

text: in

text: X

text: The

text: image

text: of

text: e∅

text: is

text: precisely

text: the

text: subspace

text: of

text: W

text: -invariants,

text: with

text: W

text: acting

text: X

text: X

text: through

text: the

text: scattering

text: operators

text: this

text: the

text: Paley–Wiener

text: theorem

text: [DHS,

text: Theorem

text: At

text: this

text: point

text: I

text: stress

text: that

text: the

text: boundary

text: degeneration

text: X∅

text: should

text: not

text: be

text: considered

text: just

text: as

text: an

text: abstract

text: G-variety,

text: but

text: as

text: a

text: G-variety

text: equipped

text: with

text: extra

text: structure,

text: which

text: is

text: obtained

text: from

text: X.

text: This

text: extra

text: structure

text: is

text: encoded

text: S

text: in

text: the

text: scattering

text: morphisms.

text: The

text: space

text: +(X∅)

text: is

text: not

text: determined

text: by

text: the

text: abstract

text: variety

text: X∅,

text: but

text: by

text: the

text: variety

text: plus

text: the

text: scattering

text: morphisms,

text: which

text: depend

text: on

text: X.

text: Now

text: consider

text: the

text: morphisms

text: induced

text: by

text: asymptotics:

text: S

text: S

text: S

text: S

text: e∅

text: e∅

text: (X)

text: (X

text: +(X∅)WX

text: +(X∅

text: )WX.

text: One

text: can

text: show

text: that

text: they

text: descend

text: to

text: morphisms

text: on

text: the

text: spaces

text: of

text: push-

text: forward

text: measures:

text: E∗

text: S

text: S

text: (X

text: X

text: /G)

text: +(X∅

text: X∅

text: /G).

text: S

text: Note

text: that

text: this

text: is

text: not

text: automatic,

text: because

text: +(X∅

text: X∅

text: /G)

text: cannot

text: be

text: iden-

text: S

text: tiﬁed

text: with

text: the

text: Gdiag-coinvariants

text: of

text: +(X∅

text: X∅).

text: For

text: example,

text: when

text: X∅

text: X∅

text: N

text: SL

text: so

text: X∅

text: X∅

text: /G

text: N

text: G/N,

text: the

text: map

text: C

text: is

text: the

text: projection

text: to

text: a

text: coordinate,

text: but

text: every

text: evaluation

text: on

text: a

text: point

text: in

text: the

text: preimage

text: of

text: C

text: is

text: an

text: N-invariant

text: distribution.

text: Nonethe-

text: less,

text: using

text: the

text: invariance

text: of

text: the

text: image

text: of

text: the

text: asymptotics

text: morphism

text: e∅

text: un-

text: E∗

text: der

text: the

text: scattering

text: operators,

text: one

text: can

text: show

text: that

text: the

text: map

text: descending

text: from

text: e∅

text: e∅

text: does

text: exist,

text: uniquely,

text: and

text: one

text: can

text: compute

text: the

text: pull-backs

text: of

text: relative

text: characters

text: under

text: this

text: map.

text: We

text: will

text: only

text: present

text: the

text: ﬁnal

text: result

text: for

text: the

text: ex-

text: amples

text: considered

text: in

text: this

text: paper.

text: Note

text: that

text: the

text: boundary

text: degeneration

text: X∅

text: for

text: X

text: is

text: isomorphic

text: to

text: A

text: P(X)

text: X

text: G,

text: whereas

text: the

text: boundary

text: degeneration

text: for

text: X

text: is

text: A

text: P(X)

text: G.

text: Thus,

text: the

text: X


text: 34 YIANNIS SAKELLARIDIS T of relative characters. Thus, in the end, the “correct” operators ∅ do not depend just on the varieties X∅, Y∅ as abstract varieties, which might be isomorphic to each other, but also on the scattering morphisms that they are endowed with (which underlie the calculation of Theorem 5.1). More precisely, for the diagram (5.10) to commute, we would need the  following to hold, using the scalars µ (χ), µ (χ) of Theorem 5.1: X Y T 1 1 µ (χ)( ∅f)(χδ 2 ) = µ (χ)fˇ(χδ 2 ). (5.11) Y P(Y ) X P(X) T Formally, at least, this means that the operator ∅ is given as multiplica- tive convolution on A by the measure ν → whose Mellin transform X X∅ Y∅ is − 1 µ (χδ 2 ) νˇX∅→Y∅(χ) = Y P−(1Y ) . (5.12) µ (χδ 2 ) X P(X) We will explicate these transfer operators case-by-case in the examples that follow. Part 2. Special examples 6. KUZNETSOV TO TRACE FORMULA FOR SL 2 Here we discuss the local comparison behind Rudnick’s thesis [Rud90]. The goal is to compare stable orbital integrals for the trace formula and the Kunzetsov formula for the group SL . Both the group G = SL , and the \ 2 2 Whittaker space (N, ψ) G have the same dual group, namely, PGL . So, 2 we expect a local transfer map T S− \ → S G : ((N, ψ) G/(N, ψ)) ( ), G which gives rise to stable functoriality between the Kuznetsov and the Sel- S− \ berg trace formula. The non-standard space ((N, ψ) G/(N, ψ)) of orbital (cid:12) integrals is the one correspon(cid:18)ding (cid:19)to L(Ad, 1), as we will recall below. \ G As before, we choose the identiﬁcation N G N = given by a (cid:18) a b (cid:55)→(cid:19) ζ := c, c d − − and the section ζ (cid:55)→ ζ˜ = ζ 1 over G . We ﬁx the cho(cid:12)sen self- ζ m G (cid:39) G (cid:39) \ dual (with respect to ψ) Haar measure on N and N G N; the a a integration formula (3.5) now reads: ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

font: JSSNYA+URWPalladioL-Roma

text: YIANNIS

text: SAKELLARIDIS

text: T

text: of

text: relative

text: characters.

text: Thus,

text: in

text: the

text: end,

text: the

text: “correct”

text: operators

text: do

text: not

text: depend

text: just

text: on

text: the

text: varieties

text: X∅,

text: Y∅

text: as

text: abstract

text: varieties,

text: which

text: might

text: be

text: isomorphic

text: to

text: each

text: other,

text: but

text: also

text: on

text: the

text: scattering

text: morphisms

text: that

text: they

text: are

text: endowed

text: with

text: (which

text: underlie

text: the

text: calculation

text: of

text: Theorem

text: More

text: precisely,

text: for

text: the

text: diagram

text: to

text: commute,

text: we

text: would

text: need

text: the

text: following

text: to

text: hold,

text: using

text: the

text: scalars

text: µ

text: (χ),

text: µ

text: (χ)

text: of

text: Theorem

text: X

text: Y

text: T

text: µ

text: (χ)(

text: ∅f)(χδ

text: µ

text: (χ)fˇ(χδ

text: Y

text: P(Y

text: X

text: P(X)

text: T

text: Formally,

text: at

text: least,

text: this

text: means

text: that

text: the

text: operator

text: is

text: given

text: as

text: multiplica-

text: tive

text: convolution

text: on

text: A

text: by

text: the

text: measure

text: ν

text: whose

text: Mellin

text: transform

text: X

text: X∅

text: Y∅

text: is

text: µ

text: (χδ

text: νˇX∅→Y∅(χ)

text: Y

text: µ

text: (χδ

text: X

text: P(X)

text: We

text: will

text: explicate

text: these

text: transfer

text: operators

text: case-by-case

text: in

text: the

text: examples

text: that

text: follow.

text: Part

text: Special

text: examples

text: KUZNETSOV

text: TO

text: TRACE

text: FORMULA

text: FOR

text: SL

text: Here

text: we

text: discuss

text: the

text: local

text: comparison

text: behind

text: Rudnick’s

text: thesis

text: The

text: goal

text: is

text: to

text: compare

text: stable

text: orbital

text: integrals

text: for

text: the

text: trace

text: formula

text: and

text: the

text: Kunzetsov

text: formula

text: for

text: the

text: group

text: SL

text: Both

text: the

text: group

text: G

text: SL

text: and

text: the

text: Whittaker

text: space

text: (N,

text: ψ)

text: G

text: have

text: the

text: same

text: dual

text: group,

text: namely,

text: PGL

text: So,

text: we

text: expect

text: a

text: local

text: transfer

text: map

text: T

text: S−

text: S

text: G

text: ((N,

text: ψ)

text: G/(N,

text: ψ))

text: G

text: which

text: gives

text: rise

text: to

text: stable

text: functoriality

text: between

text: the

text: Kuznetsov

text: and

text: the

text: Sel-

text: S−

text: berg

text: trace

text: formula.

text: The

text: non-standard

text: space

text: ((N,

text: ψ)

text: G/(N,

text: ψ))

text: of

text: orbital

text: integrals

text: is

text: the

text: one

text: L(Ad,

text: as

text: we

text: will

text: recall

text: below.

text: G

text: As

text: before,

text: we

text: choose

text: the

text: identiﬁcation

text: N

text: G

text: N

text: given

text: by

text: a

text: a

text: b

text: ζ

text: c,

text: c

text: d

text: and

text: the

text: section

text: ζ

text: ζ˜

text: ζ

text: over

text: G

text: We

text: ﬁx

text: the

text: self-

text: ζ

text: m

text: G

text: G

text: dual

text: (with

text: respect

text: to

text: ψ)

text: Haar

text: measure

text: on

text: N

text: and

text: N

text: G

text: N;

text: the

text: a

text: a

text: integration

text: formula

text: now

text: reads:

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 36 YIANNIS SAKELLARIDIS § — I point the reader to [LM15, Theorem 3.2] or [SV, 18.1] for details. The formula conjecturally generalizes to arbitrary groups, according to a con- jecture of Lapid and Mao [LM15] which resembles the Ichino–Ikeda conjec- ture [II10]. Returning to local notation, with π an irreducible generic represenation of G(F ), it is natural to demand that the correctly normalized Bessel dis- tribution J (or J , where Π denotes the L-packet of π) be such that the π Π ⊗ adjoint of (6.2), composed with eva(cid:90)luation on the cosets of 1, is the (N, ψ) − (N, ψ 1)-equivariant functional ∗ ⊗ (cid:51) ⊗ (cid:55)→ (cid:104) (cid:105) π˜ ˆ π v˜ v π(n)v, v˜ ψ(n)dn. (cid:90) (cid:90) (cid:88) N Explicitly, ∗ ⊗ (cid:104) (cid:105) J (Φ Φ ) = Φ (x )Φ (x ) π(nx )v, π˜(x )v˜ ψ(n)dn, Π 1 2 \ 1 1 2 2 1 2 (N G)2 N (v,v˜) where (v, v˜) runs over dual pairs in dual bases of π and π˜. S \ We are now seeking the transfer operator from ((N, ψ) G/(N, ψ)) to S (G) that will pull back the stable character Θ to J . If we believe that G S Π Π this operator should indeed have image in (G), which happens to be iden- S G tiﬁed with the space of coinvariants of (G) under the PGL -adjoint action, 2 then it is completely determined by this property, over all tempered pack- ets Π, since stable tempered characters are dense in the space of stable tem- pered distributions. The following theorem will be proven in [Sakd]: (cid:12) \ (cid:39) G Theorem 6.1. Fix the isomorphism N G N as before, thus identifying a § the subvariety Y of anti-diagonal matrices used to trivialize orbital integrals ( 1) G G (cid:39) G with . Fix the isomorphism via the trace map. Consider the equivari- m a T F G ant Fourier transform := of multiplicative convolution with the measure | | × Id,G1 D = ψ(ζ)dζ = ψ(ζ) ζ d ζ on . 1 m Then: S \ (1) The convolution makes sense on (N, ψ G/N, ψ) as the Fourier trans- S form of a distribution, and maps it into (G). G (2) For every tempered packet Π and J as above, Π T ∗ Θ = J . (6.4) Π Π (3) The transform extends to an isomorphism, given by the same convolution understood, again, as the Fourier transform of a(n L2-)distribution: T S− \ −→∼ S G : (N, ψ G/N, ψ) ( ), (6.5) L(Ad,1) G ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: YIANNIS

text: SAKELLARIDIS

text: I

text: point

text: the

text: reader

text: to

text: Theorem

text: or

text: [SV,

text: for

text: details.

text: The

text: formula

text: conjecturally

text: generalizes

text: to

text: arbitrary

text: groups,

text: according

text: to

text: a

text: con-

text: jecture

text: of

text: Lapid

text: and

text: Mao

text: which

text: resembles

text: the

text: Ichino–Ikeda

text: conjec-

text: ture

text: Returning

text: to

text: local

text: notation,

text: with

text: π

text: an

text: irreducible

text: generic

text: represenation

text: of

text: G(F

text: it

text: is

text: natural

text: to

text: demand

text: that

text: the

text: correctly

text: normalized

text: Bessel

text: dis-

text: tribution

text: J

text: (or

text: J

text: where

text: Π

text: denotes

text: the

text: L-packet

text: of

text: π)

text: be

text: such

text: that

text: the

text: π

text: Π

text: adjoint

text: of

text: composed

text: with

text: on

text: the

text: cosets

text: of

text: is

text: the

text: (N,

text: ψ)

text: (N,

text: ψ

text: functional

text: π˜

text: ˆ

text: π

text: v˜

text: v

text: π(n)v,

text: v˜

text: ψ(n)dn.

text: N

text: Explicitly,

text: J

text: (Φ

text: Φ

text: Φ

text: (x

text: )Φ

text: (x

text: π(nx

text: )v,

text: π˜(x

text: )v˜

text: ψ(n)dn,

text: Π

text: (N

text: N

text: (v,v˜)

text: where

text: (v,

text: v˜)

text: runs

text: over

text: dual

text: pairs

text: in

text: dual

text: bases

text: of

text: π

text: and

text: π˜.

text: S

text: We

text: are

text: now

text: seeking

text: the

text: transfer

text: operator

text: from

text: ((N,

text: ψ)

text: G/(N,

text: ψ))

text: to

text: S

text: (G)

text: that

text: will

text: pull

text: back

text: the

text: stable

text: character

text: Θ

text: to

text: J

text: If

text: we

text: believe

text: that

text: G

text: S

text: Π

text: Π

text: this

text: operator

text: should

text: indeed

text: have

text: image

text: in

text: (G),

text: which

text: happens

text: to

text: be

text: iden-

text: S

text: G

text: tiﬁed

text: with

text: the

text: space

text: of

text: coinvariants

text: of

text: (G)

text: under

text: the

text: PGL

text: -adjoint

text: action,

text: then

text: it

text: is

text: completely

text: determined

text: by

text: this

text: property,

text: over

text: all

text: tempered

text: pack-

text: ets

text: Π,

text: since

text: stable

text: tempered

text: characters

text: are

text: dense

text: in

text: the

text: space

text: of

text: stable

text: tem-

text: pered

text: distributions.

text: The

text: following

text: theorem

text: will

text: be

text: proven

text: in

text: [Sakd]:

text: G

text: Theorem

text: Fix

text: the

text: isomorphism

text: N

text: G

text: N

text: as

text: before,

text: thus

text: identifying

text: a

text: the

text: subvariety

text: Y

text: of

text: anti-diagonal

text: matrices

text: used

text: to

text: trivialize

text: orbital

text: integrals

text: G

text: G

text: G

text: with

text: Fix

text: the

text: isomorphism

text: via

text: the

text: trace

text: map.

text: Consider

text: the

text: equivari-

text: m

text: a

text: T

text: F

text: G

text: ant

text: Fourier

text: transform

text: of

text: multiplicative

text: convolution

text: with

text: the

text: measure

text: D

text: ψ(ζ)dζ

text: ψ(ζ)

text: ζ

text: d

text: ζ

text: on

text: m

text: Then:

text: S

text: The

text: convolution

text: makes

text: sense

text: on

text: (N,

text: ψ

text: G/N,

text: ψ)

text: as

text: the

text: Fourier

text: trans-

text: S

text: form

text: of

text: a

text: distribution,

text: and

text: maps

text: it

text: into

text: (G).

text: G

text: For

text: every

text: tempered

text: packet

text: Π

text: and

text: J

text: as

text: above,

text: Π

text: T

text: Θ

text: J

text: Π

text: Π

text: The

text: transform

text: extends

text: to

text: an

text: isomorphism,

text: given

text: by

text: the

text: same

text: convolution

text: understood,

text: again,

text: as

text: the

text: Fourier

text: transform

text: of

text: a(n

text: T

text: S−

text: S

text: G

text: (N,

text: ψ

text: G/N,

text: ψ)

text: G

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 38 YIANNIS SAKELLARIDIS × of 2 2 matrices of rank one, while for X = the Whittaker model we have \ X∅ = the same space N G with t}rivial character on N. E}quation (5.11) together with Theorem 5.1 in this case read: − − − T − − T fˇ(χ) = γ(χδ 12 , αˇ, 0, ψ 1) ∅f(χ) = γ(χ, αˇ, 1, ψ 1) ∅f(χ), T and Proposition 2.1 implies that the transfer operator ∅ in this case should F − − − be given by . (Note that γ(χ, αˇ, 1, ψ 1) 1 =(cid:12) γ(χ, αˇ, 0, ψ).) αˇ,1 \ G We notice that, for the coordinates on N G N and ﬁxed here, the T G T transfer operator is exactly the same as the transfer operator ∅, namely, convolution by the measure D ! 1 7. KUZNETSOV TO RELATIVE TRACE FORMULA FOR TORIC PERIODS \ (cid:39) G Now consider the case of X = T G/T , where G = PGL and T is 2 m a split torus. One could also consider a non-split torus, but would need to § slightly modify the equivariant Fourier transforms that we presented in 1. The role of the character Θ , here, will be played by a relative charac- Π ter I for an irreducible tempered representation of PGL , for the quotient π G \ G 2 space PGL / . The deﬁnition of the relative character I is com- m 2 m π pletely analogous to that of the Kuznetsov relative character J : It is given π as the composition S × → ⊗ → C (X X) π ˆ π˜ \ ⊗ (where X = T G), where the dual of the map to π ˆ π˜ is the morphism ⊗ → ∞ × π˜ π C (X X) (cid:90) × that, composed with evaluation at T 1 T 1 is given by: ⊗ (cid:55)→ (cid:104) (cid:105) v˜ v π(t)v, v˜ dt. T Here the integral is convergent (for tempered representations), and no nor- malization is needed. Moreover, the L-packets for the group PGL are sin- 2 gletons (if we do not consider its inner forms, which we should have, in the case of a non-split torus), therefore there is no need to distinguish, no- tationally, between π and its L-packet Π. The following theorem is a reformulation of results proven in [Sak13, Saka]. (cid:12) (cid:12) \ (cid:39) G Theorem 7.1. Fix the isomorphism N G N as before, and an isomorphism \ G a ∈ G T G T = such that the identity element of G maps to 1 , and the non- a a trivial element of the Weyl group of T maps to 0. T F ◦ F Consider the equivariant Fourier transform := of multiplicative • | • | ×• Id,1 GId,1 convolution, twice, with D = ψ( ) d on measures on . 1 m Then: ##LTLine## ##LTLine##

text: YIANNIS

text: SAKELLARIDIS

text: of

text: matrices

text: of

text: rank

text: one,

text: while

text: for

text: X

text: the

text: Whittaker

text: model

text: we

text: have

text: X∅

text: the

text: same

text: space

text: N

text: G

text: with

text: t}rivial

text: character

text: on

text: N.

text: E}quation

text: together

text: with

text: Theorem

text: in

text: this

text: case

text: read:

text: T

text: T

text: fˇ(χ)

text: γ(χδ

text: αˇ,

text: ψ

text: ∅f(χ)

text: γ(χ,

text: αˇ,

text: ψ

text: ∅f(χ),

text: T

text: and

text: Proposition

text: implies

text: that

text: the

text: transfer

text: operator

text: in

text: this

text: case

text: should

text: F

text: be

text: given

text: by

text: (Note

text: that

text: γ(χ,

text: αˇ,

text: ψ

text: γ(χ,

text: αˇ,

text: ψ).)

text: αˇ,1

text: G

text: We

text: notice

text: that,

text: for

text: the

text: coordinates

text: on

text: N

text: G

text: N

text: and

text: ﬁxed

text: here,

text: the

text: T

text: G

text: T

text: transfer

text: operator

text: is

text: exactly

text: the

text: same

text: as

text: the

text: transfer

text: operator

text: namely,

text: convolution

text: by

text: the

text: measure

text: D

text: KUZNETSOV

text: TO

text: RELATIVE

text: TRACE

text: FORMULA

text: FOR

text: TORIC

text: PERIODS

text: G

text: Now

text: consider

text: the

text: case

text: of

text: X

text: T

text: G/T

text: where

text: G

text: PGL

text: and

text: T

text: is

text: m

text: a

text: split

text: torus.

text: One

text: could

text: also

text: consider

text: a

text: non-split

text: torus,

text: but

text: would

text: need

text: to

text: slightly

text: modify

text: the

text: equivariant

text: Fourier

text: transforms

text: that

text: we

text: presented

text: in

text: The

text: role

text: of

text: the

text: character

text: Θ

text: here,

text: will

text: be

text: played

text: by

text: a

text: relative

text: charac-

text: Π

text: ter

text: I

text: for

text: an

text: irreducible

text: tempered

text: representation

text: of

text: PGL

text: for

text: the

text: quotient

text: π

text: G

text: G

text: space

text: PGL

text: The

text: deﬁnition

text: of

text: the

text: relative

text: character

text: I

text: is

text: com-

text: m

text: m

text: π

text: pletely

text: analogous

text: to

text: that

text: of

text: the

text: Kuznetsov

text: relative

text: character

text: J

text: It

text: is

text: given

text: π

text: as

text: the

text: composition

text: S

text: C

text: (X

text: X)

text: π

text: ˆ

text: π˜

text: (where

text: X

text: T

text: G),

text: where

text: the

text: dual

text: of

text: the

text: map

text: to

text: π

text: ˆ

text: π˜

text: is

text: the

text: morphism

text: π˜

text: π

text: C

text: (X

text: X)

text: that,

text: composed

text: with

text: evaluation

text: at

text: T

text: T

text: is

text: given

text: by:

text: v˜

text: v

text: π(t)v,

text: v˜

text: dt.

text: T

text: Here

text: the

text: integral

text: is

text: convergent

text: (for

text: tempered

text: representations),

text: and

text: no

text: nor-

text: malization

text: is

text: needed.

text: Moreover,

text: the

text: L-packets

text: for

text: the

text: group

text: PGL

text: are

text: sin-

text: gletons

text: (if

text: we

text: do

text: not

text: consider

text: its

text: inner

text: forms,

text: which

text: we

text: should

text: have,

text: in

text: the

text: case

text: of

text: a

text: non-split

text: torus),

text: therefore

text: there

text: is

text: no

text: need

text: to

text: distinguish,

text: no-

text: tationally,

text: between

text: π

text: and

text: its

text: L-packet

text: Π.

text: The

text: following

text: theorem

text: is

text: a

text: reformulation

text: of

text: results

text: proven

text: in

text: Saka].

text: G

text: Theorem

text: Fix

text: the

text: isomorphism

text: N

text: G

text: N

text: as

text: before,

text: and

text: an

text: isomorphism

text: G

text: a

text: G

text: T

text: G

text: T

text: such

text: that

text: the

text: identity

text: element

text: of

text: G

text: maps

text: to

text: and

text: the

text: non-

text: a

text: a

text: trivial

text: element

text: of

text: the

text: Weyl

text: group

text: of

text: T

text: maps

text: to

text: T

text: F

text: F

text: Consider

text: the

text: equivariant

text: Fourier

text: transform

text: of

text: multiplicative

text: convolution,

text: twice,

text: with

text: D

text: ψ(

text: d

text: on

text: measures

text: on

text: m

text: Then:

text: ##LTLine##

text: ##LTLine##


text: 46 YIANNIS SAKELLARIDIS [DHS] Patrick Delorme, Pascale Harinck, and Yiannis Sakellaridis. Paley-Wiener the- orems for a p-adic spherical variety. To appear in Memoirs of the AMS, 107pp. arXiv:1410.2279. [GJ72] Roger Godement and Herve´ Jacquet. Zeta functions of simple algebras. Lecture Notes in Mathematics, Vol. 260. Springer-Verlag, Berlin-New York, 1972. [Her12] P. Edward Herman. The functional equation and beyond endoscopy. Paciﬁc J. Math., 260(2):497–513, 2012. doi:10.2140/pjm.2012.260.497. [II10] Atsushi Ichino and Tamutsu Ikeda. On the periods of automorphic forms on special orthogonal groups and the Gross-Prasad conjecture. Geom. Funct. Anal., 19(5):1378–1425, 2010. doi:10.1007/s00039-009-0040-4. [Jac03] Herve´ Jacquet. Smooth transfer of Kloosterman integrals. Duke Math. J., 120(1):121–152, 2003. doi:10.1215/S0012-7094-03-12015-3. [Joh] Daniel Johnstone. A Gelfand–Graev formula and stable transfer factors for SL (f). Preprint. arXiv:1611.06291. n [KMSW] Tasho Kaletha, Alberto Minguez, Sug Woo Shin, and Paul-James White. Endo- scopic classiﬁcation of representations: Inner forms of unitary groups. Preprint. arXiv:1409.3731. [KS] Friedrich Knop and Barbara Schalke. The dual group of a spherical variety. To ap- pear in Transactions of the Moscow Mathematical Society. arXiv:1702.08264. [Laf14] Laurent Lafforgue. Noyaux du transfert automorphe de Langlands et for- mules de Poisson non line´aires. Jpn. J. Math., 9(1):1–68, 2014. doi:10.1007/ s11537-014-1274-y. [Lan04] Robert P. Langlands. Beyond endoscopy. In Contributions to automorphic forms, ge- ometry, and number theory, pages 611–697. Johns Hopkins Univ. Press, Baltimore, MD, 2004. [Lan13] Robert P. Langlands. Singularite´s et transfert. Ann. Math. Que´., 37(2):173–253, 2013. doi:10.1007/s40316-013-0008-5. [LM15] Erez Lapid and Zhengyu Mao. A conjecture on Whittaker-Fourier coefﬁcients of cusp forms. J. Number Theory, 146:448–505, 2015. doi:10.1016/j.jnt.2013. 10.003. [Mok15] Chung Pang Mok. Endoscopic classiﬁcation of representations of quasi-split unitary groups. Mem. Amer. Math. Soc., 235(1108):vi+248, 2015. doi:10.1090/ memo/1108. [Ngoˆ] Bao Chaˆu Ngoˆ. Geometry of arc spaces, generalized Hankel transforms and Lang- lands functoriality. Preprint. [Ngoˆ14] Bao Chaˆu Ngoˆ. On a certain sum of automorphic L-functions. In Automorphic forms and related geometry: assessing the legacy of I. I. Piatetski-Shapiro, volume 614 of Contemp. Math., pages 337–343. Amer. Math. Soc., Providence, RI, 2014. doi: 10.1090/conm/614/12270. [Pop86] V. L. Popov. Contractions of actions of reductive algebraic groups. Mat. Sb. (N.S.), 130(172)(3):310–334, 431, 1986. [Rud90] Zeev Rudnick. Poincare series. PhD thesis, 1990. Yale University. [Saka] Yiannis Sakellaridis. Beyond endoscopy for the relative trace formula II: global theory. To appear in J. Inst. Math. Jussieu, published online, 2017; 101pp. doi: 10.1017/S1474748017000032. [Sakb] Yiannis Sakellaridis. Functorial transfer between relative trace formulas in rank one. In preparation. [Sakc] Yiannis Sakellaridis. The Selberg trace formula revisited. Preprint, 80pp. arXiv: 1710.01866. [Sakd] Yiannis Sakellaridis. Transfer operators and Hankel transforms between relative trace formulas. In preparation.

text: YIANNIS

text: SAKELLARIDIS

text: [DHS]

label: reference

text: Patrick

text: Delorme,

text: Pascale

text: Harinck,

text: and

text: Yiannis

text: Sakellaridis.

text: Paley-Wiener

text: the-

text: orems

text: for

text: a

text: p-adic

text: spherical

text: variety.

text: To

text: appear

text: in

text: Memoirs

font: ZZYUIZ+URWPalladioL-Ital

text: of

text: the

text: AMS,

font: BXIXQW+NimbusMonL-Regu

text: Roger

text: Godement

text: and

text: Herve´

text: Jacquet.

text: Zeta

text: functions

text: of

text: simple

text: algebras.

text: Lecture

text: Notes

text: in

text: Mathematics,

text: Vol.

text: Springer-Verlag,

text: Berlin-New

text: York,

text: P.

text: Edward

text: Herman.

text: The

text: functional

text: equation

text: and

text: beyond

text: endoscopy.

text: Paciﬁc

text: J.

text: Math.,

text: Atsushi

text: Ichino

text: and

text: Tamutsu

text: Ikeda.

text: On

text: the

text: periods

text: of

text: automorphic

text: forms

text: on

text: special

text: orthogonal

text: groups

text: and

text: the

text: Gross-Prasad

text: conjecture.

text: Geom.

text: Funct.

text: Anal.,

text: Herve´

text: Jacquet.

text: Smooth

text: transfer

text: of

text: Kloosterman

text: integrals.

text: Duke

text: Math.

text: J.,

text: [Joh]

text: Daniel

text: Johnstone.

text: A

text: Gelfand–Graev

text: formula

text: and

text: stable

text: transfer

text: factors

text: for

text: SL

text: (f).

text: Preprint.

text: n

text: [KMSW]

text: Tasho

text: Kaletha,

text: Alberto

text: Minguez,

text: Sug

text: Woo

text: Shin,

text: and

text: Paul-James

text: White.

text: Endo-

text: scopic

text: classiﬁcation

text: of

text: representations:

text: Inner

text: forms

text: of

text: unitary

text: groups.

text: Preprint.

text: [KS]

text: Friedrich

text: Knop

text: and

text: Barbara

text: Schalke.

text: The

text: dual

text: group

text: of

text: a

text: spherical

text: variety.

text: To

text: ap-

text: pear

text: in

text: Transactions

text: of

text: the

text: Moscow

text: Mathematical

text: Society.

text: Laurent

text: Lafforgue.

text: Noyaux

text: du

text: transfert

text: automorphe

text: de

text: Langlands

text: et

text: for-

text: mules

text: de

text: Poisson

text: non

text: line´aires.

text: Jpn.

text: J.

text: Math.,

text: Robert

text: P.

text: Langlands.

text: Beyond

text: endoscopy.

text: In

text: Contributions

text: to

text: automorphic

text: forms,

text: ge-

text: ometry,

text: and

text: number

text: theory,

text: pages

text: Johns

text: Hopkins

text: Univ.

text: Press,

text: Baltimore,

text: MD,

text: Robert

text: P.

text: Langlands.

text: Singularite´s

text: et

text: transfert.

text: Ann.

text: Math.

text: Que´.,

text: Erez

text: Lapid

text: and

text: Zhengyu

text: Mao.

text: A

text: conjecture

text: on

text: Whittaker-Fourier

text: coefﬁcients

text: of

text: cusp

text: forms.

text: J.

text: Number

text: Theory,

text: Chung

text: Pang

text: Mok.

text: Endoscopic

text: classiﬁcation

text: of

text: representations

text: of

text: quasi-split

text: unitary

text: groups.

text: Mem.

text: Amer.

text: Math.

text: Soc.,

text: [Ngoˆ]

text: Bao

text: Chaˆu

text: Ngoˆ.

text: Geometry

text: of

text: arc

text: spaces,

text: generalized

text: Hankel

text: transforms

text: and

text: Lang-

text: lands

text: functoriality.

text: Preprint.

text: Bao

text: Chaˆu

text: Ngoˆ.

text: On

text: a

text: certain

text: sum

text: of

text: automorphic

text: L-functions.

text: In

text: Automorphic

text: forms

text: and

text: related

text: geometry:

text: assessing

text: the

text: legacy

text: of

text: I.

text: I.

text: Piatetski-Shapiro,

text: volume

text: of

text: Contemp.

text: Math.,

text: pages

text: Amer.

text: Math.

text: Soc.,

text: Providence,

text: RI,

text: doi:

text: V.

text: L.

text: Popov.

text: Contractions

text: of

text: actions

text: of

text: reductive

text: algebraic

text: groups.

text: Mat.

text: Sb.

text: (N.S.),

text: Zeev

text: Rudnick.

text: Poincare

text: series.

text: PhD

text: thesis,

text: Yale

text: University.

text: [Saka]

text: Yiannis

text: Sakellaridis.

text: Beyond

text: endoscopy

text: for

text: the

text: relative

text: trace

text: formula

text: II:

text: global

text: theory.

text: To

text: appear

text: in

text: J.

text: Inst.

text: Math.

text: Jussieu,

text: published

text: online,

text: doi:

text: [Sakb]

text: Yiannis

text: Sakellaridis.

text: Functorial

text: transfer

text: between

text: relative

text: trace

text: formulas

text: in

text: rank

text: one.

text: In

text: preparation.

text: [Sakc]

text: Yiannis

text: Sakellaridis.

text: The

text: Selberg

text: trace

text: formula

text: revisited.

text: Preprint,

text: arXiv:

text: [Sakd]

text: Yiannis

text: Sakellaridis.

text: Transfer

text: operators

text: and

text: Hankel

text: transforms

text: between

text: relative

text: trace

text: formulas.

text: In

text: preparation.


text: 8 YIANNIS SAKELLARIDIS by the desire to have global comparisons between relative trace formulas, so let us say a few things about the relative trace formula. The presentation that follows is quite naive, pretending that all integrals converge. This will rarely be true; instead, integrals have to be regularized. A general process § of regularization was explained in [Sak16, 5–6], but it does not cover all cases (notably, it does not cover the Arthur–Selberg trace formula). Thus, a relative trace formula in the generality presented here has not been devel- oped yet. A Let k be a global ﬁeld, its ring of adeles. Assume that X is a smooth, k quasi-afﬁne G-variety deﬁned over k. Having deﬁned the Schwartz spaces of X over any local completion k of k, we now endow them with a basic v vector at (almost all) non-Archimedean places, which is typically the char- acteristic function of X(o ) (where o is the ring of integers of k , and we as- v v v sume an integral model outside of a ﬁnite set of places), times a G-invariant measure (which should be chosen to factorize a canonical global choice of \ measure, the Tamagawa measure). We keep allowing the case X = N G, A endowed with an adele class character ψ of N( ). In the case of “extended k Schwartz spaces” mentioned above, this basic vector will need to be mod- § iﬁed — I point again to 1. Having the basic vectors, we can deﬁne global (cid:79) Schwartz spaces as restricted tensor products with respect to these vectors: (cid:48) S A S (X( )) = (X(k )). k v v × S Correspondingly, for X = [X X/G] we deﬁne the basic vector of (X(k )) S × v as the push-forward of the basic vector of (X X(k )), and the global S A v Schwartz space (X( )) as the corresponding restricted tensor product k over all places. The relative trace formula of X is a global analog of the local Plancherel for- mula. The local Plancherel formula is a decomposition of the inner product of two measures(cid:90)(ﬁxing an invariant m(cid:90) easure dx on X(F )) in terms of local relative characters: Φ (x)Φ (x) (cid:98) ⊗ 1 2 = J (Φ Φ )µ (π). (1.4) π 1 2 X dx X(F) G We recall that a relative character J on X = X(F ), associated to an admis- π S × sible representation π, is a G-invariant functional on (X X) that factors as S × → ⊗ −(cid:104)−,→(cid:105) C (X X) π ˆ π˜ . × In the group case (X = H, G = H H), this is the same as a character, if × we identify the space of G-invariant generalized functions on H H with conjugation-invariant generalized functions on H. Now, globally, we deﬁne a distribution that resembles the local inner product of two functions/measures on X, but factors through the space of automorphic functions: ##LTLine##

text: YIANNIS

text: SAKELLARIDIS

text: by

text: the

text: desire

text: to

text: have

text: global

text: comparisons

text: between

text: relative

text: trace

text: formulas,

text: so

text: let

text: us

text: say

text: a

text: few

text: things

text: about

text: the

text: relative

text: trace

text: formula.

text: The

text: presentation

text: that

text: follows

text: is

text: quite

text: naive,

text: pretending

text: that

text: all

text: integrals

text: converge.

text: This

text: will

text: rarely

text: be

text: true;

text: instead,

text: integrals

text: have

text: to

text: be

text: regularized.

text: A

text: general

text: process

text: of

text: regularization

text: was

text: explained

text: in

text: but

text: it

text: does

text: not

text: cover

text: all

text: cases

text: (notably,

text: it

text: does

text: not

text: cover

text: the

text: Arthur–Selberg

text: trace

text: formula).

text: Thus,

text: a

text: relative

text: trace

text: formula

text: in

text: the

text: generality

text: presented

text: here

text: has

text: not

text: been

text: devel-

text: oped

text: yet.

text: A

text: Let

text: k

text: be

text: a

text: global

text: ﬁeld,

text: its

text: ring

text: of

text: adeles.

text: Assume

text: that

text: X

text: is

text: a

text: smooth,

text: k

text: quasi-afﬁne

text: G-variety

text: deﬁned

text: over

text: k.

text: Having

text: deﬁned

text: the

text: Schwartz

text: spaces

text: of

text: X

text: over

text: any

text: local

text: completion

text: k

text: of

text: k,

text: we

text: now

text: endow

text: them

text: with

text: a

text: basic

text: v

text: vector

text: at

text: (almost

text: all)

text: non-Archimedean

text: places,

text: which

text: is

text: typically

text: the

text: char-

text: acteristic

text: function

text: of

text: X(o

text: (where

text: o

text: is

text: the

text: ring

text: of

text: integers

text: of

text: k

text: and

text: we

text: as-

text: v

text: v

text: v

text: sume

text: an

text: integral

text: model

text: outside

text: of

text: a

text: ﬁnite

text: set

text: of

text: places),

text: times

text: a

text: G-invariant

text: measure

text: (which

text: should

text: be

text: chosen

text: to

text: factorize

text: a

text: canonical

text: global

text: choice

text: of

text: measure,

text: the

text: Tamagawa

text: measure).

text: We

text: keep

text: allowing

text: the

text: case

text: X

text: N

text: G,

text: A

text: endowed

text: with

text: an

text: adele

text: class

text: character

text: ψ

text: of

text: N(

text: In

text: the

text: case

text: of

text: “extended

text: k

text: Schwartz

text: spaces”

text: mentioned

text: above,

text: this

text: basic

text: vector

text: will

text: need

text: to

text: be

text: mod-

text: iﬁed

text: I

text: point

text: again

text: to

text: Having

text: the

text: basic

text: vectors,

text: we

text: can

text: deﬁne

text: global

text: Schwartz

text: spaces

text: as

text: restricted

text: tensor

text: products

text: with

text: respect

text: to

text: these

text: vectors:

text: S

text: A

text: S

text: (X(

text: (X(k

text: k

text: v

text: v

text: S

text: Correspondingly,

text: for

text: X

text: [X

text: X/G]

text: we

text: deﬁne

text: the

text: basic

text: vector

text: of

text: (X(k

text: S

text: v

text: as

text: the

text: push-forward

text: of

text: the

text: basic

text: vector

text: of

text: (X

text: X(k

text: and

text: the

text: global

text: S

text: A

text: v

text: Schwartz

text: space

text: (X(

text: as

text: the

text: corresponding

text: restricted

text: tensor

text: product

text: k

text: over

text: all

text: places.

text: The

text: relative

text: trace

text: formula

text: of

text: X

text: is

text: a

text: global

text: analog

text: of

text: the

text: local

text: Plancherel

text: for-

text: mula.

text: The

text: local

text: Plancherel

text: formula

text: is

text: a

text: decomposition

text: of

text: the

text: inner

text: product

text: of

text: two

text: an

text: invariant

text: easure

text: dx

text: on

text: X(F

text: in

text: terms

text: of

text: local

text: relative

text: characters:

text: Φ

text: (x)Φ

text: (x)

text: J

text: (Φ

text: Φ

text: )µ

text: (π).

text: π

text: X

text: dx

text: X(F)

text: G

text: We

text: recall

text: that

text: a

text: relative

text: character

text: J

text: on

text: X

text: X(F

text: associated

text: to

text: an

text: admis-

text: π

text: S

text: sible

text: representation

text: π,

text: is

text: a

text: G-invariant

text: functional

text: on

text: (X

text: X)

text: that

text: factors

text: as

text: S

text: C

text: (X

text: X)

text: π

text: ˆ

text: π˜

text: In

text: the

text: group

text: case

text: (X

text: H,

text: G

text: H

text: H),

text: this

text: is

text: the

text: same

text: as

text: a

text: character,

text: if

text: we

text: identify

text: the

text: space

text: of

text: G-invariant

text: generalized

text: functions

text: on

text: H

text: H

text: with

text: conjugation-invariant

text: generalized

text: functions

text: on

text: H.

text: Now,

text: globally,

text: we

text: deﬁne

text: a

text: distribution

text: that

text: resembles

text: the

text: local

text: inner

text: product

text: of

text: two

text: functions/measures

text: on

text: X,

text: but

text: factors

text: through

text: the

text: space

text: of

text: automorphic

text: functions:

text: ##LTLine##


text: 10 YIANNIS SAKELLARIDIS S spaces (X(k )) of “stable orbital integrals” of the various completions of v § k. I point the reader to [Sak16, 6.4] for an attempt at a general formulation, (cid:88) where the relative trace formula is presented as a sum of the form: ev , (1.5) ξ ∈ ξ Xss(k) of “evaluations” at the “semisimple” global points of the stack X. Here, we will be concerned with ways to “match” the local Schwartz spaces for × different quotients of the form [X X/G]. Now let X, Y be two such quotients, corresponding to spherical varieties (cid:48) X, Y (possibly for different groups G, G , and possibly endowed with line bundles as before). The functoriality predicted by a homomorphism → LY LX (1.6) between their L-groups should be realized, locally, by a linear “transfer” map between the Schwartz spaces T S → S : (X) (Y), × × (cid:48) where X = [X X/G], Y = [Y Y/G ], satisfying certain local and global desiderata, that we now describe. T The transfer operator should be such that, locally, the pull-backs of stable tempered relative characters on Y should be stable tempered rela- tive characters on X, corresponding to the map of L-packets that should be associated to (1.6). In the cases that we consider, for an irreducible π the S × → ⊗ space of morphisms (X X) π ˆ π˜ is at most one-dimensional, so the relative character for an irreducible representation is determined uniquely up to scalar. Unlike the group case, there is no canonical normalization of this scalar, but the local factor J of the Ichino–Ikeda conjecture and its π generalizations provides a distinguished such normalization. This normal- ization will be discussed individually in some examples presented in this paper; for the general case, I point the reader to Chapter 17 of [SV]. We should also have a “fundamental lemma”, stating that the image of S S the basic vector of (X) under the transfer is the basic vector of (Y) (at non-Archimedean places), so that the transfer operator translates to a trans- fer between global Schwartz spaces: T S A → S A : (X( )) (Y( )), k k A where denotes the ring of adeles of k. k Globally, when (1.6) is an isomorphism, the map should commute with the (stable) “relative trace formula” distributions, thus we should have a

text: YIANNIS

text: SAKELLARIDIS

text: S

text: spaces

text: (X(k

text: of

text: “stable

text: orbital

text: integrals”

text: of

text: the

text: various

text: completions

text: of

text: v

text: k.

text: I

text: point

text: the

text: reader

text: to

text: for

text: an

text: attempt

text: at

text: a

text: general

text: formulation,

label: equation

text: where

text: the

text: relative

text: trace

text: formula

text: is

text: presented

text: as

text: a

text: sum

text: of

text: the

text: form:

text: ev

text: ξ

text: ξ

text: Xss(k)

text: of

text: “evaluations”

text: at

text: the

text: “semisimple”

text: global

text: points

text: of

text: the

text: stack

text: X.

text: Here,

text: we

text: will

text: be

text: concerned

text: with

text: ways

text: to

text: “match”

text: the

text: local

text: Schwartz

text: spaces

text: for

text: different

text: quotients

text: of

text: the

text: form

text: [X

text: X/G].

text: Now

text: let

text: X,

text: Y

text: be

text: two

text: such

text: quotients,

text: corresponding

text: to

text: spherical

text: varieties

text: X,

text: Y

text: (possibly

text: for

text: different

text: groups

text: G,

text: G

text: and

text: possibly

text: endowed

text: with

text: line

text: bundles

text: as

text: before).

text: The

text: functoriality

text: predicted

text: by

text: a

text: homomorphism

text: LY

text: LX

text: between

text: their

text: L-groups

text: should

text: be

text: realized,

text: locally,

text: by

text: a

text: linear

text: “transfer”

text: map

text: between

text: the

text: Schwartz

text: spaces

text: T

text: S

text: S

text: (X)

text: (Y),

text: where

text: X

text: [X

text: X/G],

text: Y

text: [Y

text: Y/G

text: satisfying

text: certain

text: local

text: and

text: global

text: desiderata,

text: that

text: we

text: now

text: describe.

text: T

text: The

text: transfer

text: operator

text: should

text: be

text: such

text: that,

text: locally,

text: the

text: pull-backs

text: of

text: stable

text: tempered

text: relative

text: characters

text: on

text: Y

text: should

text: be

text: stable

text: tempered

text: rela-

text: tive

text: characters

text: on

text: X,

text: corresponding

text: to

text: the

text: map

text: of

text: L-packets

text: that

text: should

text: be

text: associated

text: to

text: In

text: the

text: cases

text: that

text: we

text: consider,

text: for

text: an

text: irreducible

text: π

text: the

text: S

text: space

text: of

text: morphisms

text: (X

text: X)

text: π

text: ˆ

text: π˜

text: is

text: at

text: most

text: one-dimensional,

text: so

text: the

text: relative

text: character

text: for

text: an

text: irreducible

text: representation

text: is

text: determined

text: uniquely

text: up

text: to

text: scalar.

text: Unlike

text: the

text: group

text: case,

text: there

text: is

text: no

text: canonical

text: normalization

text: of

text: this

text: scalar,

text: but

text: the

text: local

text: factor

text: J

text: of

text: the

text: Ichino–Ikeda

text: conjecture

text: and

text: its

text: π

text: generalizations

text: provides

text: a

text: distinguished

text: such

text: normalization.

text: This

text: normal-

text: ization

text: will

text: be

text: discussed

text: individually

text: in

text: some

text: examples

text: presented

text: in

text: this

text: paper;

text: for

text: the

text: general

text: case,

text: I

text: point

text: the

text: reader

text: to

text: Chapter

text: of

text: [SV].

text: We

text: should

text: also

text: have

text: a

text: “fundamental

text: lemma”,

text: stating

text: that

text: the

text: image

text: of

text: S

text: S

text: the

text: basic

text: vector

text: of

text: (X)

text: under

text: the

text: transfer

text: is

text: the

text: basic

text: vector

text: of

text: (Y)

text: (at

text: non-Archimedean

text: places),

text: so

text: that

text: the

text: transfer

text: operator

text: translates

text: to

text: a

text: trans-

text: fer

text: between

text: global

text: Schwartz

text: spaces:

text: T

text: S

text: A

text: S

text: A

text: (X(

text: (Y(

text: k

text: k

text: A

text: where

text: denotes

text: the

text: ring

text: of

text: adeles

text: of

text: k.

text: k

text: Globally,

text: when

text: is

text: an

text: isomorphism,

text: the

text: map

text: should

text: commute

text: with

text: the

text: (stable)

text: “relative

text: trace

text: formula”

text: distributions,

text: thus

text: we

text: should

text: have

text: a


text: 2 Sofía A. Cora et al. fraction of galaxies in the red sequence increases as a function gardless of which physical mechanism may actually be responsible of local environmental density (Baldry et al. 2006), especially at for it, has been standard ingredient in SAMs. This eﬀect is gener- high stellar masses. However, the environmental dependence of the ally referred to as strangulation (Larson, Tinsley & Caldwell 1980; morphology fractions is milder than that of colour fractions for low Balogh, Navarro & Morris 2000). Although this simple prescription masses (Bamford et al. 2009). Outliers of these two broad colour for halo stripping can account for the presence of passive galaxies branches include galaxies in a so-called intermediate green valley. in clusters at high redshift, it results in fractions of red satellite These galaxies are interpreted as experiencing a decrease of their galaxies which are higher than those observed in groups of galaxies star-formationrate(SFR)(Schawinskietal.2014;Powelletal.2017; (e.g. Baldry et al. 2006; Weinmann et al. 2006; Kimm et al. 2009); Bait et al. 2017; Coenda et al. 2018). Star formation (SF) quenching this excess of red satellites constitutes the satellite overquenching is higher for more massive galaxies, and increases at ﬁxed stellar problem. Based on these results, there is now general consensus that mass with environmental density (given either by local density or the instantaneous stripping of the hot gas of satellites due to shock- halo mass) as becomes evident from the strong correlations of the heating is a crude approximation of the process. This is additionally fractions of quiescent galaxies with stellar mass, environment and supported by the observational evidence that large fractions of near- halo-centric radius, found both locally (Wetzel et al. 2012) and at IR-bright, early-type galaxies in groups (Jeltema et al. 2008) and high redshift (Peng et al. 2010; Muzzin et al. 2012; Lin et al. 2014; also in clusters (Sun et al. 2007; Wagner et al. 2017) have extended Jian et al. 2017; Kawinwanichakij et al. 2017). Besides, the way in X-ray emission, indicating that they retain signiﬁcant hot gas haloes which red and quiescent fractions vary with environment and stellar even in these dense environments. Studies based on smoothed par- mass are diﬀerent for central and satellite galaxies (Weinmann et al. ticle hydrodynamics (SPH) simulations have found that the hot gas 2006; Wetzel et al. 2012; Kovač et al. 2014; Darvish et al. 2017; haloes of satellites are not stripped instantly (McCarthy et al. 2008), Smethurst et al. 2017; Wang et al. 2017). although RPS of halo gas is much more eﬀective than that of disc Central and satellite galaxies of a given stellar mass are subject gas (Bekki 2009). Moreover, the eﬀect of RPS on the hot corona to very diﬀerent processes. Central galaxies are believed to reside at seems to be even milder than expected because of turbulence in the centres of their host dark matter (DM) haloes. They can receive the interface with the intracluster medium (ICM) and fuelling of new gas via cooling ﬂows, and they cannibalize satellite galaxies gas from this component to the satellite hot gas halo (Quilis et al. whose orbits in the group potential decay due to dynamical friction. 2017). Dry mergers (both minor and major) can explain the shallower This mounting evidence motivated several authors to include slope of the massive end of the red sequence (Jiménez et al. 2011). prescriptions for a gradual removal of hot halo gas in SAMs, with On the other hand, major mergers contribute to produce massive, the inherent limitations of this numerical technique. Some of them passive galaxies (van der Wel et al. 2009), doubling the mass of the massive brightest cluster galaxies since ∼ 1 (e.g. Shankar only consider the eﬀect of RPS (e.g. Font et al. 2008; Weinmann z et al. 2010; Gonzalez-Perez et al. 2014) while others also take into et al. 2015). While orbiting in the main group, satellite galaxies are account the inﬂuence of TS (Guo et al. 2011; Kimm et al. 2011; also aﬀected by galaxy encounters that may result in morphological Henriques et al. 2017; Stevens & Brown 2017). If hot gas removal transformations (Kannan et al. 2015). However, the key processes proceeds gradually by the action of RPS and/or TS, the time-scale that are believed to quench the SF in satellite galaxies are triggered of SF quenching becomes longer and the discrepancies in the frac- by gas disruption due to tidal stripping (TS, Merritt 1983) and ram tion of red/quiescent galaxies between models and observations are pressure stripping (RPS, Gunn & Gott 1972; Abadi et al. 1999). alleviated although some disagreements remain. The prescriptions During TS, the material is pulled from a galaxy by the global adopted are generally based on the analytic formulation of Mc- tidal ﬁeld of the host halo or a larger neighbour galaxy, the former Carthy et al. (2008). The eﬀect of RPS on the cold gas of galaxy being more relevant than the latter in low-velocity close encounters discs has been implemented only in few SAMs (Okamoto & Na- taking place in galaxy groups (Villalobos et al. 2014). Tidal shocks gashima 2003; Lanzoni et al. 2005; Brüggen & De Lucia 2008; at the pericentres of the orbit of satellite galaxies induce dynamical Tecce et al. 2010; Stevens & Brown 2017). The combination of instabilities and impulsive tidal heating of the stellar distribution suppressed cooling ﬂows due to strangulation and RPS of cold disc that are particularly relevant for dwarf galaxies in environments gas as in Tecce et al. (2010) worsens the problem of excess of red similar to our Local Group (Kazantzidis et al. 2011). Besides, the satellites. The presence of hot gas halo during gradual removal acts impact of TS (from negligible eﬀect to complete galaxy disruption) as a shield that helps to prevent a too eﬃcient action of RPS on also depends on the satellite morphology (Chang et al. 2013). the cold gas phase. Although this is taken into account by Stevens RPS is a result of the ram-pressure (RP) exerted by the hot & Brown (2017), the condition used seems to still produce larger diﬀuse gas of a group or cluster of galaxies on satellites moving amounts of removed cold gas than expected. through it at velocities that could be close to supersonic. This physi- cal process may account for the gradual removal of both the hot gas The star formation history of galaxies and, consequently, the halo and the cold gas disc (Boselli & Gavazzi 2006; Jaﬀé et al. 2013; fractions of star forming and passive galaxies, are strongly aﬀected Poggianti et al. 2017), giving place to ‘jellyﬁsh’ galaxies in cases by supernova (SN) feedback. Hirschmann et al. (2016) tested the im- of extreme RPS (Bellhouse et al. 2017) where the trailing stripped pact of diﬀerent schemes for stellar feedback and gas recycling using tails show signs of shock heating and gas compression. Hydrody- the semi-analytic model Gaea. They ﬁnd that a model character- namical simulations show that, on one hand, RP contributes to SF ized by a strong ejection coupled to a mass-loading that is explicitly quenching through gas stripping and, on the other, can also produce dependent on redshift is reasonably successful in reproducing the a temporary and moderate enhancement of SF (more likely to occur observed ‘anti-hierarchical’ trends in galaxy mass assembly, with in Milky Way-type disc galaxies) as a result of gas compression a delayed metal enrichment and a realistic present-day stellar and that takes place at pericentre passage (Bekki 2014; Yozin & Bekki gaseous metallicity. They adopt a redshift dependence of the mass- 2015; Steinhauser et al. 2016; Ruggiero & Lima Neto 2017). loading factor (ratio between the gas outﬂow rate from galaxies The instantaneous complete removal of the hot diﬀuse gas and the star formation rate) similar to the one provided by Muratov halo of a galaxy after its infall into a larger halo, that takes place re- et al. (2015), who analyse the galaxy-scale gaseous outﬂows in the MNRAS 000, 1–21 (2017)

font: QMIQPV+TeXGyreTermes-Regular

text: Sofía

font: FUWLVA+TeXGyreTermes-Italic

text: A.

text: Cora

text: et

text: al.

text: fraction

text: of

text: galaxies

text: in

text: the

text: red

text: sequence

text: increases

text: as

text: a

text: function

text: gardless

text: of

text: which

text: physical

text: mechanism

text: may

text: actually

text: be

text: responsible

text: of

text: local

text: environmental

text: density

text: (Baldry

text: et

text: al.

text: especially

text: at

text: for

text: it,

text: has

text: been

text: standard

text: ingredient

text: in

text: SAMs.

text: This

text: eﬀect

text: is

text: gener-

text: high

text: stellar

text: masses.

text: However,

text: the

text: environmental

text: dependence

text: of

text: the

text: ally

text: referred

text: to

text: as

text: strangulation

text: (Larson,

text: Tinsley

text: Caldwell

text: morphology

text: fractions

text: is

text: milder

text: than

text: that

text: of

text: colour

text: fractions

text: for

text: low

text: Balogh,

text: Navarro

text: Morris

text: Although

text: this

text: simple

text: prescription

text: masses

text: (Bamford

text: et

text: al.

text: Outliers

text: of

text: these

text: two

text: broad

text: colour

text: for

text: halo

text: stripping

text: can

text: account

text: for

text: the

text: presence

text: of

text: passive

text: galaxies

text: branches

text: include

text: galaxies

text: in

text: a

text: so-called

text: intermediate

text: green

text: valley.

text: in

text: clusters

text: at

text: high

text: redshift,

text: it

text: results

text: in

text: fractions

text: of

text: red

text: satellite

text: These

text: galaxies

text: are

text: interpreted

text: as

text: experiencing

text: a

text: decrease

text: of

text: their

text: galaxies

text: which

text: are

text: higher

text: than

text: those

text: observed

text: in

text: groups

text: of

text: galaxies

text: (e.g.

text: Baldry

text: et

text: al.

text: Weinmann

text: et

text: al.

text: Kimm

text: et

text: al.

text: Bait

text: et

text: al.

text: Coenda

text: et

text: al.

text: Star

text: formation

text: (SF)

text: quenching

text: this

text: excess

text: of

text: red

text: satellites

text: constitutes

text: the

text: satellite

text: overquenching

text: is

text: higher

text: for

text: more

text: massive

text: galaxies,

text: and

text: increases

text: at

text: ﬁxed

text: stellar

text: problem.

text: Based

text: on

text: these

text: results,

text: there

text: is

text: now

text: general

text: consensus

text: that

text: mass

text: with

text: environmental

text: density

text: (given

text: either

text: by

text: local

text: density

text: or

text: the

text: instantaneous

text: stripping

text: of

text: the

text: hot

text: gas

text: of

text: satellites

text: due

text: to

text: shock-

text: halo

text: mass)

text: as

text: becomes

text: evident

text: from

text: the

text: strong

text: correlations

text: of

text: the

text: heating

text: is

text: a

text: crude

text: approximation

text: of

text: the

text: process.

text: This

text: is

text: additionally

text: fractions

text: of

text: quiescent

text: galaxies

text: with

text: stellar

text: mass,

text: environment

text: and

text: supported

text: by

text: the

text: observational

text: evidence

text: that

text: large

text: fractions

text: of

text: near-

text: halo-centric

text: radius,

text: found

text: both

text: locally

text: (Wetzel

text: et

text: al.

text: and

text: at

text: IR-bright,

text: early-type

text: galaxies

text: in

text: groups

text: (Jeltema

text: et

text: al.

text: and

text: high

text: redshift

text: (Peng

text: et

text: al.

text: Muzzin

text: et

text: al.

text: Lin

text: et

text: al.

text: also

text: in

text: clusters

text: (Sun

text: et

text: al.

text: Wagner

text: et

text: al.

text: have

text: extended

text: Jian

text: et

text: al.

text: Kawinwanichakij

text: et

text: al.

text: Besides,

text: the

text: way

text: in

text: X-ray

text: emission,

text: indicating

text: that

text: they

text: retain

text: signiﬁcant

text: hot

text: gas

text: haloes

text: which

text: red

text: and

text: quiescent

text: fractions

text: vary

text: with

text: environment

text: and

text: stellar

text: even

text: in

text: these

text: dense

text: environments.

text: Studies

text: based

text: on

text: smoothed

text: par-

text: mass

text: are

text: diﬀerent

text: for

text: central

text: and

text: satellite

text: galaxies

text: (Weinmann

text: et

text: al.

text: ticle

text: hydrodynamics

text: (SPH)

text: simulations

text: have

text: found

text: that

text: the

text: hot

text: gas

text: Wetzel

text: et

text: al.

text: Kovač

text: et

text: al.

text: Darvish

text: et

text: al.

text: haloes

text: of

text: satellites

text: are

text: not

text: stripped

text: instantly

text: (McCarthy

text: et

text: al.

text: Smethurst

text: et

text: al.

text: Wang

text: et

text: al.

text: although

text: RPS

text: of

text: halo

text: gas

text: is

text: much

text: more

text: eﬀective

text: than

text: that

text: of

text: disc

text: Central

text: and

text: satellite

text: galaxies

text: of

text: a

text: given

text: stellar

text: mass

text: are

text: subject

text: gas

text: (Bekki

text: Moreover,

text: the

text: eﬀect

text: of

text: RPS

text: on

text: the

text: hot

text: corona

text: to

text: very

text: diﬀerent

text: processes.

text: Central

text: galaxies

text: are

text: believed

text: to

text: reside

text: at

text: seems

text: to

text: be

text: even

text: milder

text: than

text: expected

text: because

text: of

text: turbulence

text: in

text: the

text: centres

text: of

text: their

text: host

text: dark

text: matter

text: (DM)

text: haloes.

text: They

text: can

text: receive

text: the

text: interface

text: with

text: the

text: intracluster

text: medium

text: (ICM)

text: and

text: fuelling

text: of

text: new

text: gas

text: via

text: cooling

text: ﬂows,

text: and

text: they

text: cannibalize

text: satellite

text: galaxies

text: gas

text: from

text: this

text: component

text: to

text: the

text: satellite

text: hot

text: gas

text: halo

text: (Quilis

text: et

text: al.

text: whose

text: orbits

text: in

text: the

text: group

text: potential

text: decay

text: due

text: to

text: dynamical

text: friction.

text: Dry

text: mergers

text: (both

text: minor

text: and

text: major)

text: can

text: explain

text: the

text: shallower

text: This

text: mounting

text: evidence

text: motivated

text: several

text: authors

text: to

text: include

text: slope

text: of

text: the

text: massive

text: end

text: of

text: the

text: red

text: sequence

text: (Jiménez

text: et

text: al.

text: prescriptions

text: for

text: a

text: gradual

text: removal

text: of

text: hot

text: halo

text: gas

text: in

text: SAMs,

text: with

text: On

text: the

text: other

text: hand,

text: major

text: mergers

text: contribute

text: to

text: produce

text: massive,

text: the

text: inherent

text: limitations

text: of

text: this

text: numerical

text: technique.

text: Some

text: of

text: them

text: passive

text: galaxies

text: (van

text: der

text: Wel

text: et

text: al.

text: doubling

text: the

text: mass

text: of

text: the

text: massive

text: brightest

text: cluster

text: galaxies

text: since

text: (e.g.

text: Shankar

text: only

text: consider

text: the

text: eﬀect

text: of

text: RPS

text: (e.g.

text: Font

text: et

text: al.

text: Weinmann

text: z

text: et

text: al.

text: Gonzalez-Perez

text: et

text: al.

text: while

text: others

text: also

text: take

text: into

text: et

text: al.

text: While

text: orbiting

text: in

text: the

text: main

text: group,

text: satellite

text: galaxies

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text: account

text: the

text: inﬂuence

text: of

text: TS

text: (Guo

text: et

text: al.

text: Kimm

text: et

text: al.

text: also

text: aﬀected

text: by

text: galaxy

text: encounters

text: that

text: may

text: result

text: in

text: morphological

text: Henriques

text: et

text: al.

text: Stevens

text: Brown

text: If

text: hot

text: gas

text: removal

text: transformations

text: (Kannan

text: et

text: al.

text: However,

text: the

text: key

text: processes

text: proceeds

text: gradually

text: by

text: the

text: action

text: of

text: RPS

text: and/or

text: TS,

text: the

text: time-scale

text: that

text: are

text: believed

text: to

text: quench

text: the

text: SF

text: in

text: satellite

text: galaxies

text: are

text: triggered

text: of

text: SF

text: quenching

text: becomes

text: longer

text: and

text: the

text: discrepancies

text: in

text: the

text: frac-

text: by

text: gas

text: disruption

text: due

text: to

text: tidal

text: stripping

text: (TS,

text: Merritt

text: and

text: ram

text: tion

text: of

text: red/quiescent

text: galaxies

text: between

text: models

text: and

text: observations

text: are

text: pressure

text: stripping

text: (RPS,

text: Gunn

text: Gott

text: Abadi

text: et

text: al.

text: alleviated

text: although

text: some

text: disagreements

text: remain.

text: The

text: prescriptions

text: During

text: TS,

text: the

text: material

text: is

text: pulled

text: from

text: a

text: galaxy

text: by

text: the

text: global

text: adopted

text: are

text: generally

text: based

text: on

text: the

text: analytic

text: formulation

text: of

text: Mc-

text: tidal

text: ﬁeld

text: of

text: the

text: host

text: halo

text: or

text: a

text: larger

text: neighbour

text: galaxy,

text: the

text: former

text: Carthy

text: et

text: al.

text: The

text: eﬀect

text: of

text: RPS

text: on

text: the

text: cold

text: gas

text: of

text: galaxy

text: being

text: more

text: relevant

text: than

text: the

text: latter

text: in

text: low-velocity

text: close

text: encounters

text: discs

text: has

text: been

text: implemented

text: only

text: in

text: few

text: SAMs

text: (Okamoto

text: Na-

text: taking

text: place

text: in

text: galaxy

text: groups

text: (Villalobos

text: et

text: al.

text: Tidal

text: shocks

text: gashima

text: Lanzoni

text: et

text: al.

text: Brüggen

text: De

text: Lucia

text: at

text: the

text: pericentres

text: of

text: the

text: orbit

text: of

text: satellite

text: galaxies

text: induce

text: dynamical

text: Tecce

text: et

text: al.

text: Stevens

text: Brown

text: The

text: combination

text: of

text: instabilities

text: and

text: impulsive

text: tidal

text: heating

text: of

text: the

text: stellar

text: distribution

text: suppressed

text: cooling

text: ﬂows

text: due

text: to

text: strangulation

text: and

text: RPS

text: of

text: cold

text: disc

text: that

text: are

text: particularly

text: relevant

text: for

text: dwarf

text: galaxies

text: in

text: environments

text: gas

text: as

text: in

text: Tecce

text: et

text: al.

text: worsens

text: the

text: problem

text: of

text: excess

text: of

text: red

text: similar

text: to

text: our

text: Local

text: Group

text: (Kazantzidis

text: et

text: al.

text: Besides,

text: the

text: satellites.

text: The

text: presence

text: of

text: hot

text: gas

text: halo

text: during

text: gradual

text: removal

text: acts

text: impact

text: of

text: TS

text: (from

text: negligible

text: eﬀect

text: to

text: complete

text: galaxy

text: disruption)

text: as

text: a

text: shield

text: that

text: helps

text: to

text: prevent

text: a

text: too

text: eﬃcient

text: action

text: of

text: RPS

text: on

text: also

text: depends

text: on

text: the

text: satellite

text: morphology

text: (Chang

text: et

text: al.

text: the

text: cold

text: gas

text: phase.

text: Although

text: this

text: is

text: taken

text: into

text: account

text: by

text: Stevens

text: RPS

text: is

text: a

text: result

text: of

text: the

text: ram-pressure

text: (RP)

text: exerted

text: by

text: the

text: hot

text: Brown

text: the

text: condition

text: used

text: seems

text: to

text: still

text: produce

text: larger

text: diﬀuse

text: gas

text: of

text: a

text: group

text: or

text: cluster

text: of

text: galaxies

text: on

text: satellites

text: moving

text: amounts

text: of

text: removed

text: cold

text: gas

text: than

text: expected.

text: through

text: it

text: at

text: velocities

text: that

text: could

text: be

text: close

text: to

text: supersonic.

text: This

text: physi-

text: cal

text: process

text: may

text: account

text: for

text: the

text: gradual

text: removal

text: of

text: both

text: the

text: hot

text: gas

text: The

text: star

text: formation

text: history

text: of

text: galaxies

text: and,

text: consequently,

text: the

text: halo

text: and

text: the

text: cold

text: gas

text: disc

text: (Boselli

text: Gavazzi

text: Jaﬀé

text: et

text: al.

text: fractions

text: of

text: star

text: forming

text: and

text: passive

text: galaxies,

text: are

text: strongly

text: aﬀected

text: Poggianti

text: et

text: al.

text: giving

text: place

text: to

text: ‘jellyﬁsh’

text: galaxies

text: in

text: cases

text: by

text: supernova

text: (SN)

text: feedback.

text: Hirschmann

text: et

text: al.

text: tested

text: the

text: im-

text: of

text: extreme

text: RPS

text: (Bellhouse

text: et

text: al.

text: where

text: the

text: trailing

text: stripped

text: pact

text: of

text: diﬀerent

text: schemes

text: for

text: stellar

text: feedback

text: and

text: gas

text: recycling

text: using

text: tails

text: show

text: signs

text: of

text: shock

text: heating

text: and

text: gas

text: compression.

text: Hydrody-

text: the

text: semi-analytic

text: model

text: Gaea.

text: They

text: ﬁnd

text: that

text: a

text: model

text: character-

text: namical

text: simulations

text: show

text: that,

text: on

text: one

text: hand,

text: RP

text: contributes

text: to

text: SF

text: ized

text: by

text: a

text: strong

text: ejection

text: coupled

text: to

text: a

text: mass-loading

text: that

text: is

text: explicitly

text: quenching

text: through

text: gas

text: stripping

text: and,

text: on

text: the

text: other,

text: can

text: also

text: produce

text: dependent

text: on

text: redshift

text: is

text: reasonably

text: successful

text: in

text: reproducing

text: the

text: a

text: temporary

text: and

text: moderate

text: enhancement

text: of

text: SF

text: (more

text: likely

text: to

text: occur

text: observed

text: ‘anti-hierarchical’

text: trends

text: in

text: galaxy

text: mass

text: assembly,

text: with

text: in

text: Milky

text: Way-type

text: disc

text: galaxies)

text: as

text: a

text: result

text: of

text: gas

text: compression

text: a

text: delayed

text: metal

text: enrichment

text: and

text: a

text: realistic

text: present-day

text: stellar

text: and

text: that

text: takes

text: place

text: at

text: pericentre

text: passage

text: (Bekki

text: Yozin

text: Bekki

text: gaseous

text: metallicity.

text: They

text: adopt

text: a

text: redshift

text: dependence

text: of

text: the

text: mass-

text: Steinhauser

text: et

text: al.

text: Ruggiero

text: Lima

text: Neto

text: loading

text: factor

text: (ratio

text: between

text: the

text: gas

text: outﬂow

text: rate

text: from

text: galaxies

text: The

text: instantaneous

text: complete

text: removal

text: of

text: the

text: hot

text: diﬀuse

text: gas

text: and

text: the

text: star

text: formation

text: rate)

text: similar

text: to

text: the

text: one

text: provided

text: by

text: Muratov

text: halo

text: of

text: a

text: galaxy

text: after

text: its

text: infall

text: into

text: a

text: larger

text: halo,

text: that

text: takes

text: place

text: re-

text: et

text: al.

text: who

text: analyse

text: the

text: galaxy-scale

text: gaseous

text: outﬂows

text: in

text: the

text: MNRAS


text: Semi-Analytic Galaxies I 13 this trend even more pronounced for satellites. On the other hand, as sSFR distributions† of galaxies with stellar masses within the szh(cid:38)ow1n iannFdigis. 5s,litghhetSlyFRloDweinr matozde=l s0a.gTβ1h.i3sipsahtitgehrneritshaanlsionpsraegsefonrt raanndgelosgl(oMg(cid:63)(M[M(cid:63)[(cid:12)M])(cid:12)∈]) [∈10[.95,.71,11.00.]1. ]S, altoegll(iMte(cid:63)g[aMla(cid:12)x]i)es∈ar[e10d.i1s,c1ri0m.5i-] when considering sets of galaxies within diﬀerent 0 stellar mass nated according to the mass of their main host haloes. In all cases, z = ranges. Thus the reduction in the levels of sSFR in model sag , both central and satellite galaxies have a well distinguished peak at 1 3 with the associated decreasing trend with decreasing stellar masβs,.is high values of sSFR (sSFR 10−10 yr−1). A bimodal distribution > originated in the large amount of stellar mass acquired by low mass emerges only for the most massive galaxies considered. The break galaxies as the result of the higher SF activity at higher redshifts. of the bimodality takes place at sSFR 10−10 7 yr−1, a value in = . The fact that the latter is in better agreement with observations, and good agreement with the one adopted by Brown et al. (2017) and model sag gives an excess in the low mass end of the 2 slightly higher than the one identiﬁed from the observational cata- 1 3 z = SMF (bottoβm.panel of Fig. 1) calls for a more eﬃcient model of TS logue constructed by W12. While the sSFR distribution of galaxies able to remove the excess of stellar mass in low mass galaxies, as in their observational sample exhibits a clear bimodality at all stel- already discussed in previous sections. lar masses, regardless of their classiﬁcation in centrals or satellites The diﬀerences in normalization and shape of the main se- (see their ﬁg. 1), the authors clarify that the strong sharpness of the quence of central and satellite galaxies are not explained by the peak near sSFR 10−12 yr−1 is a result of an artifact in spectral = particular treatment of environmental eﬀects introduced in sag. reductions. They also emphasize that the distributions exhibit a tail Avoiding the action of RPS on the hot and cold gas phases only pro- to much lower sSFR, like the sSFR distributions of our model galax- duces a higher number of active satellite galaxies, without changing ies do, which are in better agreement with the behaviour shown by the mean values of the sSFR of star-forming ones. In order to under- Salim et al. (2007), as it was described in Section . Consistent with stand the origin of these diﬀerences, we examine the relationship the discussion also presented in that section when analysing the between stellar mass and DM halo mass. We consider the mass of main sequence of central and satellite galaxies at 0, the high z = the main host haloes for central galaxies and the subhaloes masses sSFR peak of the distributions of satellite galaxies is slightly shifted for satellite galaxies, excluding orphans. Fig. 7 shows such relations, towards lower sSFR values than for centrals, while such a trend is representing galaxies for the calibrated model sag with a coloured not evident in the observational sample. contour map on top of which the mean values of the stellar mass DespitethesediﬀerencesbetweenourresultsandthoseofW12, for each bin of halo mass (depicted by squares) are superimposed. there are several similarities in relevant aspects of the distributions. Mean values of model sag are also added (represented by tri- As in the observational sample, more massive central galaxies have 1 3 angles), showing that the SβHM. relation is not appreciably aﬀected larger probability of having low sSFR at the expense of a reduction by a change in the parameter . Mean values are compared with the in the peak height at high sSFR. However, satellites only show such β ﬁtting function for the stellar-to-halo mass relation at 0 derived a trend for those cases where the bimodality becomes evident in z = by Moster et al. (2010); such parametrization was estimated by re- the model. Both in the simulated and observed sample, there are quiring that the observed SMF obtained from SDSS DR3 (Panter more satellites with low sSFR than centrals while the opposite oc- et al. 2004) is reproduced when populating with galaxies the haloes curs for high sSFR, regardless of the stellar mass bin considered. and subhaloes in an N-body simulation. Note that this comparison Besides, the number of satellite galaxies with low(high) sSFR in- is not fair for satellite galaxies, but it is done with the purpose of creases(decreases) systematically as we consider more massive DM highlighting the diﬀerences in the relation for centrals and satellites. haloes. These general results remain valid for galaxies generated by The general trend is very well reproduced by model central galaxies. model sag . A quantitative comparison with observational re- 1 3 For halo mass Mvir (cid:46) 1012 M(cid:12), diﬀerences between centrals and sults is attaβin.ed through the estimation of the fraction of quenched satellites become more evident. For a given stellar mass, central galaxies for diﬀerent stellar mass and halo mass ranges. galaxies inhabit more massive haloes, while satellites prefer less 6.4 Quenched fractions of central and satellite galaxies from massive ones, being the dispersion of the relation (vertical error the model SAG bars) larger for smaller stellar masses. Systematic diﬀerences in the From a given population of model galaxies selected according halo virial mass between main host and satellite haloes translate to their stellar mass and/or main host halo mass, we estimate the into diﬀerences in the hot halo mass and associated cooling rates, quenched fraction by considering those galaxies that are passive which impact on the SF activity leaving diﬀerent imprints in the at a given time of interest, that is, those that satisfy the criterion main sequence of central and satellite galaxies. sSFR 10−10 7 yr−1, as emerges from the distribution of sSFR 6.3 Speciﬁc star formation rate: distribution < . analysed in the previous section. We compare model results with We analyse the distribution of sSFR for central and satellite those obtained by W12. Although these authors consider a galaxy galaxies within diﬀerent stellar mass ranges, discriminating the lat- as passive when its sSFR 10−11 yr−1, we prefer to estimate the < ter according to the mass of their main host DM haloes . We select quenched fraction taking into account the separation between active (cid:63) galaxies within the same stellar and halo mass ranges as W12, and passive galaxies that emerge from the model at the expense of who construct galaxy group catalogues using a group-ﬁnding al- accepting a shift in the sSFR cut with respect to the one proposed gorithm based on the one presented by Yang et al. (2005). W12 by W12. Stellar mass and main host halo mass dependence of the derive stellar masses and sSFR for galaxies within groups and quenchedfractionat 0( )areshown intheleftandrightpan- z = f qz0 clusters from the spectroscopic information provided by the SDSS els of Fig. 9, respectively. In both cases, the same mass bins as those Data Release 7; the median redshift of the sample is 0 045. chosen by W12 are used. Model results are identiﬁed with lines. Er- z = . Left, middle and right panels of Fig. 8 show, respectively, the ror bars show the 68 per cent bayesian conﬁdence interval estimated From hereafter, all the sampling of satellite galaxies is done according to † Probability density function at the bin, normalized such that the integral (cid:63) the masses of their main host haloes, even if it is simply referred as halo. over the range is equal to unity. MNRAS 000, 1–21 (2017)

text: Semi-Analytic

text: Galaxies

text: I

text: this

text: trend

text: even

text: more

text: pronounced

text: for

text: satellites.

text: On

text: the

text: other

text: hand,

text: as

text: sSFR

text: distributions†

text: of

text: galaxies

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text: stellar

text: masses

text: within

text: the

text: iannFdigis.

text: matozde=l

text: ]S,

text: when

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text: the

text: mass

text: of

text: their

text: main

text: host

text: haloes.

text: In

text: all

text: cases,

text: z

text: ranges.

text: Thus

text: the

text: reduction

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text: the

text: levels

text: of

text: sSFR

text: in

text: model

text: sag

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text: central

text: and

text: satellite

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text: have

text: a

text: well

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text: the

text: associated

text: decreasing

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text: decreasing

text: stellar

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text: high

text: values

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text: sSFR

text: (sSFR

text: A

text: bimodal

text: distribution

text: originated

text: in

text: the

text: large

text: amount

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text: stellar

text: mass

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text: low

text: mass

text: emerges

text: only

text: for

text: the

text: most

text: massive

text: galaxies

text: considered.

text: The

text: break

text: galaxies

text: as

text: the

text: result

text: of

text: the

text: higher

text: SF

text: activity

text: at

text: higher

text: redshifts.

text: of

text: the

text: bimodality

text: takes

text: place

text: at

text: sSFR

text: a

text: value

text: in

text: The

text: fact

text: that

text: the

text: latter

text: is

text: in

text: better

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text: observations,

text: and

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text: Brown

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text: slightly

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text: the

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text: excess

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text: clear

text: bimodality

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text: all

text: stel-

text: already

text: discussed

text: in

text: previous

text: sections.

text: lar

text: masses,

text: regardless

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text: their

text: classiﬁcation

text: in

text: centrals

text: or

text: satellites

text: The

text: diﬀerences

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text: normalization

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text: main

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text: (see

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text: ﬁg.

text: the

text: authors

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text: the

text: strong

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text: the

text: quence

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text: central

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text: satellite

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text: not

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text: the

text: peak

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text: sSFR

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text: environmental

text: eﬀects

text: introduced

text: in

text: sag.

text: reductions.

text: They

text: also

text: emphasize

text: that

text: the

text: distributions

text: exhibit

text: a

text: tail

text: Avoiding

text: the

text: action

text: of

text: RPS

text: on

text: the

text: hot

text: and

text: cold

text: gas

text: phases

text: only

text: pro-

text: to

text: much

text: lower

text: sSFR,

text: like

text: the

text: sSFR

text: distributions

text: of

text: our

text: model

text: galax-

text: duces

text: a

text: higher

text: number

text: of

text: active

text: satellite

text: galaxies,

text: without

text: changing

text: ies

text: do,

text: which

text: are

text: in

text: better

text: agreement

text: with

text: the

text: behaviour

text: shown

text: by

text: the

text: mean

text: values

text: of

text: the

text: sSFR

text: of

text: star-forming

text: ones.

text: In

text: order

text: to

text: under-

text: Salim

text: et

text: al.

text: as

text: it

text: was

text: described

text: in

text: Section

text: Consistent

text: with

text: stand

text: the

text: origin

text: of

text: these

text: diﬀerences,

text: we

text: examine

text: the

text: relationship

text: the

text: discussion

text: also

text: presented

text: in

text: that

text: section

text: when

text: analysing

text: the

text: between

text: stellar

text: mass

text: and

text: DM

text: halo

text: mass.

text: We

text: consider

text: the

text: mass

text: of

text: main

text: sequence

text: of

text: central

text: and

text: satellite

text: galaxies

text: at

text: the

text: high

text: z

text: the

text: main

text: host

text: haloes

text: for

text: central

text: galaxies

text: and

text: the

text: subhaloes

text: masses

text: sSFR

text: peak

text: of

text: the

text: distributions

text: of

text: satellite

text: galaxies

text: is

text: slightly

text: shifted

text: for

text: satellite

text: galaxies,

text: excluding

text: orphans.

text: Fig.

text: shows

text: such

text: relations,

text: towards

text: lower

text: sSFR

text: values

text: than

text: for

text: centrals,

text: while

text: such

text: a

text: trend

text: is

text: representing

text: galaxies

text: for

text: the

text: calibrated

text: model

text: sag

text: with

text: a

text: coloured

text: not

text: evident

text: in

text: the

text: observational

text: sample.

text: contour

text: map

text: on

text: top

text: of

text: which

text: the

text: mean

text: values

text: of

text: the

text: stellar

text: mass

text: for

text: each

text: bin

text: of

text: halo

text: mass

text: (depicted

text: by

text: squares)

text: are

text: superimposed.

text: there

text: are

text: several

text: similarities

text: in

text: relevant

text: aspects

text: of

text: the

text: distributions.

text: Mean

text: values

text: of

text: model

text: sag

text: are

text: also

text: added

text: (represented

text: by

text: tri-

text: As

text: in

text: the

text: observational

text: sample,

text: more

text: massive

text: central

text: galaxies

text: have

text: angles),

text: showing

text: that

text: the

text: SβHM.

text: relation

text: is

text: not

text: appreciably

text: aﬀected

text: larger

text: probability

text: of

text: having

text: low

text: sSFR

text: at

text: the

text: expense

text: of

text: a

text: reduction

text: by

text: a

text: change

text: in

text: the

text: parameter

text: Mean

text: values

text: are

text: compared

text: with

text: the

text: in

text: the

text: peak

text: height

text: at

text: high

text: sSFR.

text: However,

text: satellites

text: only

text: show

text: such

text: β

text: ﬁtting

text: function

text: for

text: the

text: stellar-to-halo

text: mass

text: relation

text: at

text: derived

text: a

text: trend

text: for

text: those

text: cases

text: where

text: the

text: bimodality

text: becomes

text: evident

text: in

text: z

text: by

text: Moster

text: et

text: al.

text: such

text: parametrization

text: was

text: estimated

text: by

text: re-

text: the

text: model.

text: Both

text: in

text: the

text: simulated

text: and

text: observed

text: sample,

text: there

text: are

text: quiring

text: that

text: the

text: observed

text: SMF

text: obtained

text: from

text: SDSS

text: (Panter

text: more

text: satellites

text: with

text: low

text: sSFR

text: than

text: centrals

text: while

text: the

text: opposite

text: oc-

text: et

text: al.

text: is

text: reproduced

text: when

text: populating

text: with

text: galaxies

text: the

text: haloes

text: curs

text: for

text: high

text: sSFR,

text: regardless

text: of

text: the

text: stellar

text: mass

text: bin

text: considered.

text: and

text: subhaloes

text: in

text: an

text: N-body

text: simulation.

text: Note

text: that

text: this

text: comparison

text: Besides,

text: the

text: number

text: of

text: satellite

text: galaxies

text: with

text: low(high)

text: sSFR

text: in-

text: is

text: not

text: fair

text: for

text: satellite

text: galaxies,

text: but

text: it

text: is

text: done

text: with

text: the

text: purpose

text: of

text: creases(decreases)

text: systematically

text: as

text: we

text: consider

text: more

text: massive

text: DM

text: highlighting

text: the

text: diﬀerences

text: in

text: the

text: relation

text: for

text: centrals

text: and

text: satellites.

text: haloes.

text: These

text: general

text: results

text: remain

text: valid

text: for

text: galaxies

text: generated

text: by

text: The

text: general

text: trend

text: is

text: very

text: well

text: reproduced

text: by

text: model

text: central

text: galaxies.

text: model

text: sag

text: A

text: quantitative

text: comparison

text: with

text: observational

text: re-

text: For

text: halo

text: mass

text: Mvir

text: diﬀerences

text: between

text: centrals

text: and

text: sults

text: is

text: attaβin.ed

text: through

text: the

text: estimation

text: of

text: the

text: fraction

text: of

text: quenched

text: satellites

text: become

text: more

text: evident.

text: For

text: a

text: given

text: stellar

text: mass,

text: central

text: galaxies

text: for

text: diﬀerent

text: stellar

text: mass

text: and

text: halo

text: mass

text: ranges.

text: galaxies

text: inhabit

text: more

text: massive

text: haloes,

text: while

text: satellites

text: prefer

text: less

text: Quenched

text: fractions

text: of

text: central

text: and

text: satellite

text: galaxies

text: from

text: massive

text: ones,

text: being

text: the

text: dispersion

text: of

text: the

text: relation

text: (vertical

text: error

text: the

text: model

text: SAG

text: bars)

text: larger

text: for

text: smaller

text: stellar

text: masses.

text: Systematic

text: diﬀerences

text: in

text: the

text: From

text: a

text: given

text: population

text: of

text: model

text: galaxies

text: selected

text: according

text: halo

text: virial

text: mass

text: between

text: main

text: host

text: and

text: satellite

text: haloes

text: translate

text: to

text: their

text: stellar

text: mass

text: and/or

text: main

text: host

text: halo

text: mass,

text: we

text: estimate

text: the

text: into

text: diﬀerences

text: in

text: the

text: hot

text: halo

text: mass

text: and

text: associated

text: cooling

text: rates,

text: quenched

text: fraction

text: by

text: considering

text: those

text: galaxies

text: that

text: are

text: passive

text: which

text: impact

text: on

text: the

text: SF

text: activity

text: leaving

text: diﬀerent

text: imprints

text: in

text: the

text: at

text: a

text: given

text: time

text: of

text: interest,

text: that

text: is,

text: those

text: that

text: satisfy

text: the

text: criterion

text: main

text: sequence

text: of

text: central

text: and

text: satellite

text: galaxies.

text: sSFR

text: as

text: emerges

text: from

text: the

text: distribution

text: of

text: sSFR

text: Speciﬁc

text: star

text: formation

text: rate:

text: distribution

text: analysed

text: in

text: the

text: previous

text: section.

text: We

text: compare

text: model

text: results

text: with

text: We

text: analyse

text: the

text: distribution

text: of

text: sSFR

text: for

text: central

text: and

text: satellite

text: those

text: obtained

text: by

text: Although

text: these

text: authors

text: consider

text: a

text: galaxy

text: galaxies

text: within

text: diﬀerent

text: stellar

text: mass

text: ranges,

text: discriminating

text: the

text: lat-

text: as

text: passive

text: when

text: its

text: sSFR

text: we

text: prefer

text: to

text: estimate

text: the

text: ter

text: according

text: to

text: the

text: mass

text: of

text: their

text: main

text: host

text: DM

text: haloes

text: We

text: select

text: quenched

text: fraction

text: taking

text: into

text: account

text: the

text: separation

text: between

text: active

text: galaxies

text: within

text: the

text: same

text: stellar

text: and

text: halo

text: mass

text: ranges

text: as

text: and

text: passive

text: galaxies

text: that

text: emerge

text: from

text: the

text: model

text: at

text: the

text: expense

text: of

text: who

text: construct

text: galaxy

text: group

text: catalogues

text: using

text: a

text: group-ﬁnding

text: al-

text: accepting

text: a

text: shift

text: in

text: the

text: sSFR

text: cut

text: with

text: respect

text: to

text: the

text: one

text: proposed

text: gorithm

text: based

text: on

text: the

text: one

text: presented

text: by

text: Yang

text: et

text: al.

text: by

text: Stellar

text: mass

text: and

text: main

text: host

text: halo

text: mass

text: dependence

text: of

text: the

text: derive

text: stellar

text: masses

text: and

text: sSFR

text: for

text: galaxies

text: within

text: groups

text: and

text: quenchedfractionat

text: )areshown

text: intheleftandrightpan-

text: z

text: f

text: clusters

text: from

text: the

text: spectroscopic

text: information

text: provided

text: by

text: the

text: SDSS

text: els

text: of

text: Fig.

text: respectively.

text: In

text: both

text: cases,

text: the

text: same

text: mass

text: bins

text: as

text: those

text: Data

text: Release

text: the

text: median

text: redshift

text: of

text: the

text: sample

text: is

text: chosen

text: by

text: are

text: used.

text: Model

text: results

text: are

text: identiﬁed

text: with

text: lines.

text: Er-

text: z

text: Left,

text: middle

text: and

text: right

text: panels

text: of

text: Fig.

text: show,

text: respectively,

text: the

text: ror

text: bars

text: show

text: the

text: per

text: cent

text: bayesian

text: conﬁdence

text: interval

text: estimated

text: From

text: hereafter,

text: all

text: the

text: sampling

text: of

text: satellite

text: galaxies

text: is

text: done

text: according

text: to

text: Probability

text: density

text: function

text: at

text: the

text: bin,

text: normalized

text: such

text: that

text: the

text: integral

text: the

text: masses

text: of

text: their

text: main

text: host

text: haloes,

text: even

text: if

text: it

text: is

text: simply

text: referred

text: as

text: halo.

text: over

text: the

text: range

text: is

text: equal

text: to

text: unity.

text: MNRAS