text: (cid:70) 20 MOTIEJUS VALIUNAS Note that G has rational growth (with respect to ∈ X(v)) as it is linkA v linkA ≥ a special subgroup of G, and so by Theorem 1 it follows that, for all n 1, (cid:101) (cid:101) ≤ ≤ (19) C nα1λn S (n) D nα1λn (cid:101) link(cid:101)A 1 GlinkA linkA 1 ∈ Z ≥ for some D > C > 0 and some α ≥ , λ 1. Now by linkA linkA × 1 0 1 construction G(Γˆ, Hˆ ) = G K where K is a ﬁnitely generated abelian linkA group; in particular, as K and G(Γˆ, Hˆ ) have polynomial and exponential growth, respectively, the group G has exponential growth, i.e. λ > 1. linkA 1 Then (18), (19) and Lemma 20 imply that either λ > λ , or λ = λ and ≥ ≥ ≥ 1 0 1 0 α α . Thus (λ , α ) (λ , α ) (λ, α) in the lexicographical order. 1 0 1 1 0 0 ⊆ But as S (n) S (n) by Corollary 13, the equations (19) and (17) GlinkA G≥ indeed imply that (λ, α) (λ , α ) in the lexicographical order, and so 1 1 (λ , α ) = (λ , α ) = (λ, α), 1 1 0 0 (cid:101) (cid:101) i.e. ††† ≤ ≤ ( ) C nαλn S (n) D nαλn. linkA GlinkA linkA × Finally, note that the group G(cid:101) ∪ = G G is a special subgroup of A ⊆linkA A linkA ∗∗∗ ††† G and so one has BGA∪linkA(n) BG(n). It the(cid:101)n follows from ( ), ( ) and Lemma 20 that for any D > 0 one has ≥ ≥ BG(n) BG ∪ (n) Dnαλn A linkA when n is suﬃciently large, which contradicts (17). This completes the proof of Theorem 5. References 1. Y. Antol´ın, A. Martino, and E. Ventura, Degree of commutativity of inﬁnite groups, Proc. Amer. Math. Soc. 145 (2017), 479–485. 2. D. A. Barkauskas, Centralisers in graph products of groups, J. Algebra 312 (2007), no. 1, 9–32. Z 3. M. Benson, Growth series of ﬁnite extensions of n are rational, Invent. Math. 73 (1983), 251–269. 4. J. Berstel, Sur les poˆles et le quotient de Hadamard de s´eries N-rationnelles, C. R. Acad. Sci. A 272 (1971), 1079–1081. 5. A. S. Besicovitch, Almost periodic functions, Dover publications, 1954. 6. I. M. Chiswell, The growth series of a graph product, B. Lond. Math. Soc. 26 (1994), no. 3, 268–272. 7. M. Coornaert, Mesures de Patterson-Sullivan sur le bord d’un espace hyperbolique au sens de Gromov, Paciﬁc J. Math. 159 (1993), no. 2, 241–270. 8. M. Duchin and M. Shapiro, Rational growth in the Heisenberg group, preprint, avail- able at arXiv: 1411.4201v2 [math.GR], 2014. 9. P. Erdo˝s and P. Tur´an, On some problems of a statistical group-theory, IV, Acta Math. Acad. Sci. H. 19 (1968), no. 3–4, 413–435. 10. E. R. Green, Graph products of groups, Ph.D. thesis, The University of Leeds, 1990.

text: MOTIEJUS

text: VALIUNAS

text: Note

text: that

text: G

text: has

text: rational

text: growth

text: (with

text: respect

text: to

text: X(v))

text: as

text: it

text: is

text: linkA

text: v

text: linkA

text: a

text: special

text: subgroup

text: of

text: G,

text: and

text: so

text: by

text: Theorem

text: it

text: follows

text: that,

text: for

text: all

text: n

text: C

text: S

text: (n)

text: D

text: GlinkA

text: linkA

text: Z

text: for

text: some

text: D

text: C

text: and

text: some

text: α

text: λ

text: Now

text: by

text: linkA

text: linkA

text: construction

text: G(Γˆ,

text: Hˆ

text: G

text: K

text: where

text: K

text: is

text: a

text: ﬁnitely

text: generated

text: abelian

text: linkA

text: group;

text: in

text: particular,

text: as

text: K

text: and

text: G(Γˆ,

text: Hˆ

text: have

text: polynomial

text: and

text: exponential

text: growth,

text: respectively,

text: the

text: group

text: G

text: has

text: exponential

text: growth,

text: i.e.

text: λ

text: linkA

text: Then

text: and

text: Lemma

text: imply

text: that

text: either

text: λ

text: λ

text: or

text: λ

text: λ

text: and

text: α

text: α

text: Thus

text: (λ

text: α

text: (λ

text: α

text: (λ,

text: α)

text: in

text: the

text: lexicographical

text: order.

text: But

text: as

text: S

text: (n)

text: S

text: (n)

text: by

text: Corollary

text: the

text: equations

text: and

text: GlinkA

text: G≥

text: indeed

text: imply

text: that

text: (λ,

text: α)

text: (λ

text: α

text: in

text: the

text: lexicographical

text: order,

text: and

text: so

text: (λ

text: α

text: (λ

text: α

text: (λ,

text: α),

text: i.e.

text: C

text: nαλn

text: S

text: (n)

text: D

text: nαλn.

text: linkA

text: GlinkA

text: linkA

text: Finally,

text: note

text: that

text: the

text: group

text: G

text: G

text: is

text: a

text: special

text: subgroup

text: of

text: A

text: ⊆linkA

text: A

text: linkA

text: G

text: and

text: so

text: one

text: has

text: BGA∪linkA(n)

text: BG(n).

text: It

text: follows

text: from

text: and

text: Lemma

text: that

text: for

text: any

text: D

text: one

text: has

text: BG(n)

text: BG

text: (n)

text: Dnαλn

text: A

text: linkA

text: when

text: n

text: is

text: suﬃciently

text: large,

text: which

text: contradicts

text: This

text: completes

text: the

text: proof

text: of

text: Theorem

text: References

text: Y.

text: Antol´ın,

text: A.

text: Martino,

text: and

text: E.

text: Ventura,

text: Degree

text: of

text: commutativity

text: of

text: inﬁnite

text: groups,

text: Proc.

text: Amer.

text: Math.

text: Soc.

text: D.

text: A.

text: Barkauskas,

text: Centralisers

text: in

text: graph

text: products

text: of

text: groups,

text: J.

text: Algebra

text: no.

text: Z

text: M.

text: Benson,

text: Growth

text: series

text: of

text: ﬁnite

text: extensions

text: of

text: n

text: are

text: rational,

text: Invent.

text: Math.

text: J.

text: Berstel,

text: Sur

text: les

text: poˆles

text: et

text: le

text: quotient

text: de

text: Hadamard

text: de

text: s´eries

text: N-rationnelles,

text: C.

text: R.

text: Acad.

text: Sci.

text: A

text: A.

text: S.

text: Besicovitch,

text: Almost

text: periodic

text: functions,

text: Dover

text: publications,

text: I.

text: M.

text: Chiswell,

text: The

text: growth

text: series

text: of

text: a

text: graph

text: product,

text: B.

text: Lond.

text: Math.

text: Soc.

text: no.

text: M.

text: Coornaert,

text: Mesures

text: de

text: Patterson-Sullivan

text: sur

text: le

text: bord

text: d’un

text: espace

text: hyperbolique

text: au

text: sens

text: de

text: Gromov,

text: Paciﬁc

text: J.

text: Math.

text: no.

text: M.

text: Duchin

text: and

text: M.

text: Shapiro,

text: Rational

text: growth

text: in

text: the

text: Heisenberg

text: group,

text: preprint,

text: avail-

text: able

text: at

text: arXiv:

text: [math.GR],

text: P.

text: Erdo˝s

text: and

text: P.

text: Tur´an,

text: On

text: some

text: problems

text: of

text: a

text: statistical

text: group-theory,

text: IV,

text: Acta

text: Math.

text: Acad.

text: Sci.

text: H.

text: no.

text: E.

text: R.

text: Green,

text: Graph

text: products

text: of

text: groups,

text: Ph.D.

text: thesis,

text: The

text: University

text: of

text: Leeds,


text: Performance of continuous quantum thermal devices indirectly connected to ∗ environments J. Onam Gonz´alez,1,1 Daniel Alonso,1,1 and Jos´e P. Palao1,1 A general quantum thermodynamics network is composed of thermal devices connected to the en- vironments through quantum wires. The coupling between the devices and the wires may introduce additional decay channels which modify the system performance with respect to the directly-coupled device. We analyze this eﬀect in a quantum three-level device connected to a heat bath or to a work source through a two-level wire. The steady state heat currents are decomposed into the contri- butions of the set of simple circuits in the graph representing the master equation. Each circuit is associated with a mechanism in the device operation and the system performance can be described by a small number of circuit representatives of those mechanisms. Although in the limit of weak coupling between the device and the wire the new irreversible contributions can become small, they prevent the system from reaching the Carnot eﬃciency. PACS numbers: 05.30.-d, 05.70.Ln, 07.20.Pe

text: Performance

label: title

text: of

text: continuous

text: quantum

text: thermal

text: devices

text: indirectly

text: connected

text: to

label: paragraph

text: environments

text: J.

label: author

text: Onam

text: Daniel

text: and

text: Jos´e

text: P.

text: A

label: abstract

text: general

text: quantum

text: thermodynamics

text: network

text: is

text: composed

text: of

text: thermal

text: devices

text: connected

text: to

text: the

text: en-

text: vironments

text: through

text: quantum

text: wires.

text: The

text: coupling

text: between

text: the

text: devices

text: and

text: the

text: wires

text: may

text: introduce

text: additional

text: decay

text: channels

text: which

text: modify

text: the

text: system

text: performance

text: with

text: respect

text: to

text: the

text: directly-coupled

text: device.

text: We

text: analyze

text: this

text: eﬀect

text: in

text: a

text: quantum

text: three-level

text: device

text: connected

text: to

text: a

text: heat

text: bath

text: or

text: to

text: a

text: work

text: source

text: through

text: a

text: two-level

text: wire.

text: The

text: steady

text: state

text: heat

text: currents

text: are

text: decomposed

text: into

text: the

text: contri-

text: butions

text: of

text: the

text: set

text: of

text: simple

text: circuits

text: in

text: the

text: graph

text: representing

text: the

text: master

text: equation.

text: Each

text: circuit

text: is

text: associated

text: with

text: a

text: mechanism

text: in

text: the

text: device

text: operation

text: and

text: the

text: system

text: performance

text: can

text: be

text: described

text: by

text: a

text: small

text: number

text: of

text: circuit

text: representatives

text: of

text: those

text: mechanisms.

text: Although

text: in

text: the

text: limit

text: of

text: weak

text: coupling

text: between

text: the

text: device

text: and

text: the

text: wire

text: the

text: new

text: irreversible

text: contributions

text: can

text: become

text: small,

text: they

text: prevent

text: the

text: system

text: from

text: reaching

text: the

text: Carnot

text: eﬃciency.

text: PACS

text: numbers:


text: 4 (cid:80) where Wα is the transition rate from the state j to the state i due to the coupling with the bath α. In the following ij W will denote the matrix with elements W = W(cid:88)α. The diagonal elements satisfy ij α=c,w,h ij − Wα = Wα , (7) ii ji (cid:54) j=i implying the conservation of the normalization. Besides, as a consequence of the Kubo-Martin-Schwinger condition in (4), the forward and backward transition rates are relat(cid:18)ed by (cid:19) (cid:126) Wα − ω ji = exp ij . (8) Wα k T ij B α Equation (6) is the starting point of our analysis. When the system is driven by a work source we will arrive to an equation with a similar structure, and the results described below will also apply to that case. B. Circuit ﬂuxes and aﬃnities Q S Here we describe how to determine the heat currents ˙ and the entropy production ˙ in the steady state. In the α following we assume that the currents are positive when the energy ﬂows towards the system. The method is based G | | on the representation of the master equation (6) by a connected graph (V, E), being V = N the number of vertices, | | representing the system states, and E the number of undirected edges, representing the transitions between diﬀerent C G states. A simple circuit of is a closed path with no repetition of vertices or edges. Some procedures to determine ν the set of circuits in a graph are discussed in appendix A. Each one of the two possible diﬀerent orientations of a C −C simple circuit, denoted by (cid:126) and (cid:126) , is a cycle. A cycle then consists of a sequence of directed edges with transition ν ν rates Wα, and it has an associated algebraic value [25] (cid:89) ij A C A C ((cid:126) ) = α((cid:126) ) , (9) ν ν α=c,w,h with (cid:89) A C α((cid:126) ) = Wα , (10) (cid:81) ν ij ∈ ij ν C where ∈ denotes the product of all the transition rates due to the bath α in the cycle (cid:126) . If the cycle does not ij ν A C ν involve the bath α, α((cid:126) ) = 1 for consistency. ν The cycle aﬃnity [25] is deﬁned by (cid:32) (cid:33) (cid:88) A C C C ((cid:126) ) X((cid:126)ν) = Xα((cid:126)ν) = kB ln A(−C(cid:126)ν ) , (11) α=c,w,h ν (cid:80) C A C A −C where the aﬃnity associated with each bath is Xα((cid:126) ) = k ln[ α((cid:126) )/ α( (cid:126) )]. When the system is only coupled ν B ν ν with thermal baths, the same amount of energy is taken and transferred to them in a complete cycle, implying C T Xα((cid:126) ) = 0. However, when the system is in addition coupled to a work source the summation may diﬀer from α α ν zero, indicating the net exchange of energy between the work source and the baths. The cycle ﬂux is deﬁned by [30] C − − |C A C − A −C I((cid:126) ) = D 1 det( W )[ ((cid:126) ) ( (cid:126) )] , (12) (cid:102) (cid:102)ν ν ν ν | | where D = det(W) . The matrix W is obtained from the rate matrix W replacing the elements of an arbitrary row − |C − by ones, and ( W ) denotes the matrix resulting from removing from W all the rows and columns corresponding ν to the vertices of the circuit C . Considering the relation between the diagonal and non-diagonal elements of W, the ν ##LTLine## ##LTLine## ##LTLine##

text: where

text: Wα

text: is

text: the

text: transition

text: rate

text: from

text: the

text: state

text: j

text: to

text: the

text: state

text: i

text: due

text: to

text: the

text: coupling

text: with

text: the

text: bath

text: α.

text: In

text: the

text: following

text: ij

text: W

text: will

text: denote

text: the

text: matrix

text: with

text: elements

text: W

label: equation

text: The

text: diagonal

text: elements

text: satisfy

text: ij

text: α=c,w,h

text: ij

text: Wα

text: Wα

text: ii

text: ji

font: default

text: j=i

text: implying

text: the

text: conservation

text: of

text: the

text: normalization.

text: Besides,

text: as

text: a

text: consequence

text: of

text: the

text: Kubo-Martin-Schwinger

text: condition

text: in

text: the

text: forward

text: and

text: backward

text: transition

text: rates

text: are

text: by

text: Wα

text: ω

text: ji

text: exp

text: ij

text: Wα

text: k

text: T

text: ij

text: B

text: α

text: Equation

text: is

text: the

text: starting

text: point

text: of

text: our

text: analysis.

text: When

text: the

text: system

text: is

text: driven

text: by

text: a

text: work

text: source

text: we

text: will

text: arrive

text: to

text: an

text: equation

text: with

text: a

text: similar

text: structure,

text: and

text: the

text: results

text: described

text: below

text: will

text: also

text: apply

text: to

text: that

text: case.

text: B.

text: Circuit

text: ﬂuxes

text: and

text: aﬃnities

text: Q

text: S

text: Here

text: we

text: describe

text: how

text: to

text: determine

text: the

text: heat

text: currents

text: and

text: the

text: entropy

text: production

text: in

text: the

text: steady

text: state.

text: In

text: the

text: α

text: following

text: we

text: assume

text: that

text: the

text: currents

text: are

text: positive

text: when

text: the

text: energy

text: ﬂows

text: towards

text: the

text: system.

text: The

text: method

text: is

text: based

text: G

text: on

text: the

text: representation

text: of

text: the

text: master

text: equation

text: by

text: a

text: connected

text: graph

text: (V,

text: E),

text: being

text: V

text: N

text: the

text: number

text: of

text: vertices,

text: representing

text: the

text: system

text: states,

text: and

text: E

text: the

text: number

text: of

text: undirected

text: edges,

text: representing

text: the

text: transitions

text: between

text: diﬀerent

text: C

text: G

text: states.

text: A

text: simple

text: circuit

text: of

text: is

text: a

text: closed

text: path

text: with

text: no

text: repetition

text: of

text: vertices

text: or

text: edges.

text: Some

text: procedures

text: to

text: determine

text: ν

text: the

text: set

text: of

text: circuits

text: in

text: a

text: graph

text: are

text: discussed

text: in

text: appendix

text: A.

text: Each

text: one

text: of

text: the

text: two

text: possible

text: diﬀerent

text: orientations

text: of

text: a

text: C

text: −C

text: simple

text: circuit,

text: denoted

text: by

text: and

text: is

text: a

text: cycle.

text: A

text: cycle

text: then

text: consists

text: of

text: a

text: sequence

text: of

text: directed

text: edges

text: with

text: transition

text: ν

text: ν

text: rates

text: Wα,

text: and

text: it

text: has

text: an

text: associated

text: algebraic

text: value

text: ij

text: A

text: C

text: A

text: C

text: ν

text: ν

text: α=c,w,h

text: with

text: A

text: C

text: Wα

text: ν

text: ij

text: ij

text: ν

text: C

text: where

text: denotes

text: the

text: product

text: of

text: all

text: the

text: transition

text: rates

text: due

text: to

text: the

text: bath

text: α

text: in

text: the

text: cycle

text: If

text: the

text: cycle

text: does

text: not

text: ij

text: ν

text: A

text: C

text: ν

text: involve

text: the

text: bath

text: α,

text: for

text: consistency.

text: ν

text: The

text: cycle

text: aﬃnity

text: is

text: deﬁned

text: by

text: A

text: C

text: C

text: C

text: kB

text: ln

text: α=c,w,h

text: ν

text: C

text: A

text: C

text: A

text: −C

text: where

text: the

text: aﬃnity

text: associated

text: with

text: each

text: bath

text: is

text: k

text: ln[

text: α(

text: When

text: the

text: system

text: is

text: only

text: coupled

text: ν

text: B

text: ν

text: ν

text: with

text: thermal

text: baths,

text: the

text: same

text: amount

text: of

text: energy

text: is

text: taken

text: and

text: transferred

text: to

text: them

text: in

text: a

text: complete

text: cycle,

text: implying

text: C

text: T

text: However,

text: when

text: the

text: system

text: is

text: in

text: addition

text: coupled

text: to

text: a

text: work

text: source

text: the

text: summation

text: may

text: diﬀer

text: from

text: α

text: α

text: ν

text: zero,

text: indicating

text: the

text: net

text: exchange

text: of

text: energy

text: between

text: the

text: work

text: source

text: and

text: the

text: baths.

text: The

text: cycle

text: ﬂux

text: is

text: deﬁned

text: by

text: C

text: |C

text: A

text: C

text: A

text: −C

text: D

text: det(

text: W

text: ν

text: ν

text: ν

text: where

text: D

text: det(W)

text: The

text: matrix

text: W

text: is

text: obtained

text: from

text: the

text: rate

text: matrix

text: W

text: replacing

text: the

text: elements

text: of

text: an

text: arbitrary

text: row

text: |C

text: by

text: ones,

text: and

text: W

text: denotes

text: the

text: matrix

text: resulting

text: from

text: removing

text: from

text: W

text: all

text: the

text: rows

text: and

text: columns

text: corresponding

text: ν

text: to

text: the

text: vertices

text: of

text: the

text: circuit

text: C

text: Considering

text: the

text: relation

text: between

text: the

text: diagonal

text: and

text: non-diagonal

text: elements

text: of

text: W,

text: the

text: ν

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 2 JACK BUTTCANE the associated information necessary for a number theorist to apply analysis on this space. We classify the spectral parameters of the cusp forms at the minimal K-types in Theorem 4, and then, taking as input the spectral parameters and Fourier-Whittaker coeﬃcients of such minimal cusp forms, generate the rest of the spectral expansion. The main theorem is Theorem 5. We give expressions for the Whittaker functions at the minimal K-types in Theorem 6; these are due, in greatly diﬀerent language, to Miyazaki [24] who obtained them by studying their diﬀerential equations and then applying a certain induction argument, but we take this a step farther by evaluating the base case directly from the Jacquet integral, thus ﬁxing the relationship between elements of the representation and the Whittaker model, as well as the cusp forms, via the Fourier expansion. This is signiﬁcant for the theory of special functions p q on GL 3 because the Mellin-Barnes integral is simpler analytically, but the Jacquet integral has a certain algebraic structure that we rely on in constructions such as [5]. Conversely, we must then rely on Miyazaki’s solution of the diﬀerential equations (or representation theory) for the uniqueness property. A primary tenet of this paper, in combination with the previous part [6], is to be self- contained. We generally succeed here except for three pieces which are collected into Theorem 1: First, we do not prove the initial spectral decomposition; this is a standard proof using the techniques of Selberg studying integral operators ﬁrst carried out by [10] and can be found in the ﬁrst few pages of Harish-Chandra’s book [13], or Langland’s paper [19], or in a number of other books, e.g. [25], see also [15, Appendix A]. Second, we quote the trivial bound for the principal series representations; this is a bound on Langlands parameters for the conjecturally non-existentent complementary series and can be found in [28]. Lastly, we do not prove multiplicity one for Whittaker functions; from the analytic perspective, we are avoiding the solution to a lengthy problem in partial diﬀerential equations, but it follows in the representation-theoretic language from Shalika’s paper [27]. “ p q Z The proofs below will generally assume that the quotient is by Γ SL 3, , but of course only the Fourier expansion will see the particular discrete group in use. 2. The main theorems We continue in the notation of part I [6]. A A ě Let be the space of smooth scalar-valued cusp forms. Let d, d 0 be the space D P A p q “ of smooth, vector-valued cusp forms with K-type d; that is f d satisﬁes f gk p qD p q p ` q f g d k . (So in particular, f is a 2d 1 -dimensional row vector.) In Section , we A A describe the decomposition of into the spaces d by the projection operators on K. As discussed in the introduction, we will use three external results about these cusp forms. Theorem 1. (1) The space of scalar-valued cusp forms decomposes into a direct sum of simultaneous A eigenspaces of the Casimir diﬀerential operators ∆ and ∆ (see Section ). We µ 1 2 parameterize the eigenspaces by the eigenvalues of the power function “ p q p q “ ´ ` ` p q “ ∆ p ` λ µ p ` , λ µ 1 µ21 µ22 µ23 , λ µ µ µ µ . i ρ µ i ρ µ 1 2 2 1 2 3 A A A Again, the eigenspaces further decompose into d. The space d is ﬁnite- µ µ µ dimensional. ##LTLine##

text: JACK

text: BUTTCANE

text: the

text: associated

text: information

text: necessary

text: for

text: a

text: number

text: theorist

text: to

text: apply

text: analysis

text: on

text: this

text: space.

text: We

text: classify

text: the

text: spectral

text: parameters

text: of

text: the

text: cusp

text: forms

text: at

text: the

text: minimal

text: K-types

text: in

text: Theorem

text: and

text: then,

text: taking

text: as

text: input

text: the

text: spectral

text: parameters

text: and

text: Fourier-Whittaker

text: coeﬃcients

text: of

text: such

text: minimal

text: cusp

text: forms,

text: generate

text: the

text: rest

text: of

text: the

text: spectral

text: expansion.

text: The

text: main

text: theorem

text: is

text: Theorem

text: We

text: give

text: expressions

text: for

text: the

text: Whittaker

text: functions

text: at

text: the

text: minimal

text: K-types

text: in

text: Theorem

text: these

text: are

text: due,

text: in

text: greatly

text: diﬀerent

text: language,

text: to

text: Miyazaki

text: who

text: obtained

text: them

text: by

text: studying

text: their

text: diﬀerential

text: equations

text: and

text: then

text: applying

text: a

text: certain

text: induction

text: argument,

text: but

text: we

text: take

text: this

text: a

text: step

text: farther

text: by

text: evaluating

text: the

text: base

text: case

text: directly

text: from

text: the

text: Jacquet

text: integral,

text: thus

text: ﬁxing

text: the

text: relationship

text: between

text: elements

text: of

text: the

text: representation

text: and

text: the

text: Whittaker

text: model,

text: as

text: well

text: as

text: the

text: cusp

text: forms,

text: via

text: the

text: Fourier

text: expansion.

text: This

text: is

text: signiﬁcant

text: for

text: the

text: theory

text: of

text: special

text: functions

text: p

text: q

text: on

text: GL

text: because

text: the

text: Mellin-Barnes

text: integral

text: is

text: simpler

text: analytically,

text: but

text: the

text: Jacquet

text: integral

text: has

text: a

text: certain

text: algebraic

text: structure

text: that

text: we

text: rely

text: on

text: in

text: constructions

text: such

text: as

text: Conversely,

text: we

text: must

text: then

text: rely

text: on

text: Miyazaki’s

text: solution

text: of

text: the

text: diﬀerential

text: equations

text: (or

text: representation

text: theory)

text: for

text: the

text: uniqueness

text: property.

text: A

text: primary

text: tenet

text: of

text: this

text: paper,

text: in

text: combination

text: with

text: the

text: previous

text: part

text: is

text: to

text: be

text: self-

text: contained.

text: We

text: generally

text: succeed

text: here

text: except

text: for

text: three

text: pieces

text: which

text: are

text: collected

text: into

text: Theorem

text: First,

text: we

text: do

text: not

text: prove

text: the

text: initial

text: spectral

text: decomposition;

text: this

text: is

text: a

text: standard

text: proof

text: using

text: the

text: techniques

text: of

text: Selberg

text: studying

text: integral

text: operators

text: ﬁrst

text: carried

text: out

text: by

text: and

text: can

text: be

text: found

text: in

text: the

text: ﬁrst

text: few

text: pages

text: of

text: Harish-Chandra’s

text: book

text: or

text: Langland’s

text: paper

text: or

text: in

text: a

text: number

text: of

text: other

text: books,

text: e.g.

text: see

text: also

text: Appendix

text: A].

text: Second,

text: we

text: quote

text: the

text: trivial

text: bound

text: for

text: the

text: principal

text: series

text: representations;

text: this

text: is

text: a

text: bound

text: on

text: Langlands

text: parameters

text: for

text: the

text: conjecturally

text: non-existentent

text: complementary

text: series

text: and

text: can

text: be

text: found

text: in

text: Lastly,

text: we

text: do

text: not

text: prove

text: multiplicity

text: one

text: for

text: Whittaker

text: functions;

text: from

text: the

text: analytic

text: perspective,

text: we

text: are

text: avoiding

text: the

text: solution

text: to

text: a

text: lengthy

text: problem

text: in

text: partial

text: diﬀerential

text: equations,

text: but

text: it

text: follows

text: in

text: the

text: representation-theoretic

text: language

text: from

text: Shalika’s

text: paper

text: p

text: q

text: Z

text: The

text: proofs

text: below

text: will

text: generally

text: assume

text: that

text: the

text: quotient

text: is

text: by

text: Γ

text: SL

text: but

text: of

text: course

text: only

text: the

text: Fourier

text: expansion

text: will

text: see

text: the

text: particular

text: discrete

text: group

text: in

text: use.

text: The

text: main

text: theorems

text: We

text: continue

text: in

text: the

text: notation

text: of

text: part

text: I

text: A

text: A

text: ě

text: Let

text: be

text: the

text: space

text: of

text: smooth

text: scalar-valued

text: cusp

text: forms.

text: Let

text: d,

text: d

text: be

text: the

text: space

text: D

text: P

text: A

text: p

text: q

text: of

text: smooth,

text: vector-valued

text: cusp

text: forms

text: with

text: K-type

text: d;

text: that

text: is

text: f

text: d

text: satisﬁes

text: f

text: gk

text: p

text: qD

text: p

text: q

text: p

text: q

text: f

text: g

text: d

text: k

text: (So

text: in

text: particular,

text: f

text: is

text: a

text: -dimensional

text: row

text: vector.)

text: In

text: Section

text: we

text: A

text: A

text: describe

text: the

text: decomposition

text: of

text: into

text: the

text: spaces

text: d

text: by

text: the

text: projection

text: operators

text: on

text: K.

text: As

text: discussed

text: in

text: the

text: introduction,

text: we

text: will

text: use

text: three

text: external

text: results

text: about

text: these

text: cusp

text: forms.

text: Theorem

text: The

text: space

text: of

text: scalar-valued

text: cusp

text: forms

text: decomposes

text: into

text: a

text: direct

text: sum

text: of

text: simultaneous

text: A

text: eigenspaces

text: of

text: the

text: Casimir

text: diﬀerential

text: operators

text: and

text: (see

text: Section

text: We

text: µ

text: parameterize

text: the

text: eigenspaces

text: by

text: the

text: eigenvalues

text: of

text: the

text: power

text: function

text: p

text: q

text: p

text: q

text: p

text: q

text: p

text: λ

text: µ

text: p

text: λ

text: µ

text: λ

text: µ

text: µ

text: µ

text: µ

text: i

text: ρ

text: µ

text: i

text: ρ

text: µ

text: A

text: A

text: A

text: Again,

text: the

text: eigenspaces

text: further

text: decompose

text: into

text: d.

text: The

text: space

text: d

text: is

text: ﬁnite-

text: µ

text: µ

text: µ

text: dimensional.

text: ##LTLine##


text: p q HIGHER WEIGHT ON GL 3 , II: THE CUSP FORMS. 3 A ‰ t u P t u p ` ´ ` ´ q | | ă (2) If d 0 wżith d 0, 1 and µ of theżform x it, x it, 2it , then x 1. µ P A 2 (3) The n-th Fourier coeﬃcient of φ lies in the span of the Jacquet integrals. That µ is, p q p q “ p q p q φ ug ψ u du I w ug ψ u du, pZqz pRq n pRq µ,f l n p qU“ U p q p q U p q with I xyk p ` y f k for some smooth f k . The integral on the right con- µ,f ρ µ p q ą p q ą p q verges absolutely on Re µ Re µ Re µ , and must be interpretted in the 1 p q “ 2 p q 3 Riemann sense when some Re µ Re µ . i j Proof. Part (1) is Theorem 3 and Lemma 18 in [13]. Part (2) is the unitary dual estimate noted in [22, Appendix A.1]; of course, stronger “ “ estimates are known for d 0, e.g [17, Appendix 2], and the d 1 case should also follow by Rankin-Selberg theory (as [16] does for the unramiﬁed case), but to the author’s knowledge, no one has yet performed this computation. For part (3), we note that φ generates a representation of G via right translation; it has the Fr´echet topology generated from the Lie algebra. This representation is equivalent to a subrepresentation [18, Theorem 8.37] of a principal series representation t | p q “ p q p q p qu f f xyvk p ` y χ v f k ρ µ for some µ and χ. The operator on the ﬁrst representation which takes a cusp form to its n- th Fourier coeﬃcient has image in the Whittaker model, as does the Jacquet integral on the second representation, and they respect the Fr´echet toplology. Shalika’s local multiplicity one theorem [27, Theorem 3.1] states these operators (from the ﬁrst representation to the (cid:3) Whittaker model) are identical. In Section , we will show the Lie algebra gives rise to ﬁve diﬀerential-vector operators A Ñ A ` “ ´ ´ Y a : d d a, a 2, 1, 0, 1, 2 which are left-translation invariant. We describe the cusp forms by their behavior under the action of these operators. In Sections and , we will show ` ˘ ` ˘ Theorem 2. Deﬁne the diﬀerential operator “ ` ` ´ ` ´ P (1) Λ 27∆2 4 ∆ x2 1 ∆ 4x2 1 2 , x R. x 2 1 1 P A Then for f d, the following are equivalent: A ´ A ´ (1) f is orthogonal to Y 1 d 1 and Y 2 d 2. ´ ´ (2) f is a zero of Y 1, Y 2 and an eigenfunction of Y 0. “ “ (3) d 0, d 1 or f is a zero of the operator Λ ´ . d 1 2 A ˚ We say that such an f is at its minimal K-type and we deﬁne d to be the space of such A ˚ forms. Further deﬁne d be the subspace of simultaneous eigenfunctions of ∆ and ∆ with µ 1 2 eigenvalues matching p ` . ρ µ The value of a description as in part (3) is that it involves only the particular K-type we A ´ A ´ are interested in, without reference to d 1 or d 2. This argument of Section also gives an equivalent statement of the Strong Selberg Eigenvalue Conjecture: ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: p

text: q

text: HIGHER

text: WEIGHT

text: ON

text: GL

text: II:

text: THE

text: CUSP

text: FORMS.

text: A

label: list

text: t

text: u

text: P

text: t

text: u

text: p

text: q

text: ă

text: If

text: d

text: wżith

text: d

text: and

text: µ

text: of

text: theżform

text: x

text: it,

text: x

text: it,

text: then

text: x

text: µ

text: P

text: A

text: The

text: n-th

text: Fourier

text: coeﬃcient

text: of

text: φ

text: lies

text: in

text: the

text: span

text: of

text: the

text: Jacquet

text: integrals.

text: That

text: µ

text: is,

text: p

text: q

text: p

text: q

text: p

text: q

text: p

text: q

text: φ

text: ug

text: ψ

text: u

text: du

text: I

text: w

text: ug

text: ψ

text: u

text: du,

text: pZqz

text: pRq

text: n

text: pRq

text: µ,f

text: l

text: n

text: p

text: qU“

text: U

text: p

text: q

text: p

text: q

text: U

text: p

text: q

text: with

text: I

text: xyk

text: p

text: y

text: f

text: k

text: for

text: some

text: smooth

text: f

text: k

text: The

text: integral

text: on

text: the

text: right

text: con-

text: µ,f

text: ρ

text: µ

text: p

text: q

text: ą

text: p

text: q

text: ą

text: p

text: q

text: verges

text: absolutely

text: on

text: Re

text: µ

text: Re

text: µ

text: Re

text: µ

text: and

text: must

text: be

text: interpretted

text: in

text: the

text: p

text: q

text: p

text: q

text: Riemann

text: sense

text: when

text: some

text: Re

text: µ

text: Re

text: µ

text: i

text: j

text: Proof.

text: Part

text: is

text: Theorem

text: and

text: Lemma

text: in

text: Part

text: is

text: the

text: unitary

text: dual

text: estimate

text: noted

text: in

text: Appendix

text: of

text: course,

text: stronger

text: estimates

text: are

text: known

text: for

text: d

text: e.g

text: Appendix

text: and

text: the

text: d

text: case

text: should

text: also

text: follow

text: by

text: Rankin-Selberg

text: theory

text: (as

text: does

text: for

text: the

text: unramiﬁed

text: case),

text: but

text: to

text: the

text: author’s

text: knowledge,

text: no

text: one

text: has

text: yet

text: performed

text: this

text: computation.

text: For

text: part

text: we

text: note

text: that

text: φ

text: generates

text: a

text: representation

text: of

text: G

text: via

text: right

text: translation;

text: it

text: has

text: the

text: Fr´echet

text: topology

text: generated

text: from

text: the

text: Lie

text: algebra.

text: This

text: representation

text: is

text: equivalent

text: to

text: a

text: subrepresentation

text: Theorem

text: of

text: a

text: principal

text: series

text: representation

text: t

text: p

text: q

text: p

text: q

text: p

text: q

text: p

text: qu

text: f

text: f

text: xyvk

text: p

text: y

text: χ

text: v

text: f

text: k

text: ρ

text: µ

text: for

text: some

text: µ

text: and

text: χ.

text: The

text: operator

text: on

text: the

text: ﬁrst

text: representation

text: which

text: takes

text: a

text: cusp

text: form

text: to

text: its

text: n-

text: th

text: Fourier

text: coeﬃcient

text: has

text: image

text: in

text: the

text: Whittaker

text: model,

text: as

text: does

text: the

text: Jacquet

text: integral

text: on

text: the

text: second

text: representation,

text: and

text: they

text: respect

text: the

text: Fr´echet

text: toplology.

text: Shalika’s

text: local

text: multiplicity

text: one

text: theorem

text: Theorem

text: states

text: these

text: operators

text: (from

text: the

text: ﬁrst

text: representation

text: to

text: the

text: Whittaker

text: model)

text: are

text: identical.

text: In

text: Section

text: we

text: will

text: show

text: the

text: Lie

text: algebra

text: gives

text: rise

text: to

text: ﬁve

text: diﬀerential-vector

text: operators

text: A

text: Ñ

text: A

text: Y

text: a

text: d

text: d

text: a,

text: a

text: which

text: are

text: left-translation

text: invariant.

text: We

text: describe

text: the

text: cusp

text: forms

text: by

text: their

text: behavior

text: under

text: the

text: action

text: of

text: these

text: operators.

text: In

text: Sections

text: and

text: we

text: will

text: show

text: Theorem

text: Deﬁne

text: the

text: diﬀerential

text: operator

text: P

text: Λ

text: x

text: R.

text: x

text: P

text: A

text: Then

text: for

text: f

text: d,

text: the

text: following

text: are

text: equivalent:

text: A

text: A

text: f

text: is

text: orthogonal

text: to

text: Y

text: d

text: and

text: Y

text: d

text: f

text: is

text: a

text: zero

text: of

text: Y

text: Y

text: and

text: an

text: eigenfunction

text: of

text: Y

text: d

text: d

text: or

text: f

text: is

text: a

text: zero

text: of

text: the

text: operator

text: Λ

text: d

text: A

text: We

text: say

text: that

text: such

text: an

text: f

text: is

text: at

text: its

text: minimal

text: K-type

text: and

text: we

text: deﬁne

text: d

text: to

text: be

text: the

text: space

text: of

text: such

text: A

text: forms.

text: Further

text: deﬁne

text: d

text: be

text: the

text: subspace

text: of

text: simultaneous

text: eigenfunctions

text: of

text: and

text: with

text: µ

text: eigenvalues

text: matching

text: p

text: ρ

text: µ

text: The

text: value

text: of

text: a

text: description

text: as

text: in

text: part

text: is

text: that

text: it

text: involves

text: only

text: the

text: particular

text: K-type

text: we

text: A

text: A

text: are

text: interested

text: in,

text: without

text: reference

text: to

text: d

text: or

text: d

text: This

text: argument

text: of

text: Section

text: also

text: gives

text: an

text: equivalent

text: statement

text: of

text: the

text: Strong

text: Selberg

text: Eigenvalue

text: Conjecture:

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: 32 JACK BUTTCANE p q p q ÞÑ p´ ´ q and T d w , µw2 is diagonal and satisﬁes a symmetry relation under m, m m, m , 2 so it suﬃces to compute ˘D p q p ` qD p q ˘ D p q p ` qD p q ud, d w ΓdW u, 1 d w , ud,´ d w ΓdW u, 1 d w . d l l d 1 l l There is a coinc˜iden`ce o˘f form among the m`atrice˘s in the functi¸onal equations of the Whittaker functions, the cons`tant terms o˘f the Eisenstein` series and t˘he Whittaker functions ` ˘ ` ˘ at a degenerate character, and we use that here. It is easy to check that ` ` p ` q “ ´ Γ 1 u ` ` Γ 1 u ` R p´ q W p ´ q ΓdW u, 1 π 12 ´2 Σd`` Σd`´ i ´ 2 Σd´´ Σd´` d i d 0, u . Γ u Γ 1 u 2 2 So to compute D p q p ż` qD p q ˆ ˆ ˙˙ d w ΓdW u, 1 d w , l l r it is suﬃcient to compute 8 ` p q “ D p qR p´ q W p ´ qD p q “ p ` q´ ` D ?x i F d u : d w3 d i d 0, u dż w3 ´8 1 x2 12 u d k 1, 1 ` x2 , 1 dx, which has components p q “ 1 p ´ q´ ´ p q F d 1 u : 1 x2 1 u2 dd 1 x dx, m ,m ´ m ,m 1 We may deduce from the symmetries (24) that p q “ p¯ q ` p q p q “ p¯ q ´ ` p q (61) F˘d u 1 d mF˘d u , F˘dp ´ q u 1 d 1 mF˘d ´ u . d,m dm,d d 1 ,m ` ˘ ` m˘,d 1 ´ p q ` ˘ ` ˘ For the moment, we assume Re u is large. From (23), (25), and [12, 3.196.3], we have p q ´ 1´ ` 1´ p q “ ´ 2d ! Γ d m u Γ d m u F d 1 u 2 dπ 12 p ` 1q p ´ 1q 2´ ` ´2 . m ,d ż d m ! d m ! Γ d u Γ d 1 u 2 2 “ ´b p ´ ´ 1 ´ ` 1qp q “ ´ 1 When m d 1, (25) becomes P d 1 m ,d 1 m x dx m , so 1 p q F d 1 ´ u “ 1żp ´ q´ ` ´ ´ 1´ p ` q´ ` ` ` 1´ ´ m ,d p1 ´ q d ´ 1 x 1 d 1 2m u 1 x 1 d 1 2m u dx 21 d p ` 21dq p1´! 1q 1 d m ! d m ! ż ´ 1 p ´ q´ ` ´ ´ 1´ p ` q´ ` ´ ` 1´ d 1 x 1 d 1 2m u 1 x 1 d 1 2m u dx ´ 1 ` ˘ ` ˘ ` ˘ ´ 1 1 p ´ q´ ` ´ ´ 1´ p ` q´ ` ´ ` 1´ m 1 `x 1˘ d `1 2m u ˘1 ` x 1 ˘d 1 2m u dx ´ 1 ´ ´ 1´ ´ ` 1´ ´ “ ´ 1 Γ d 1 m u Γ d 1 m u Γ 1 u m π 12 2´ ` ´ 2 ´ ` 2 . Γ d u Γ d 1 u Γ u 1 2 2 2 “ p´ q If ε 1 d, then ˘D p q “ ˘ ´ ˘ ˘ D p q “ ˘ ´ ¯ ud, d w i dud, ε, ud,´ d w i1 dud,´ ε. d 2 d d 1 2 d 1 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: JACK

text: BUTTCANE

text: p

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text: p

text: q

text: ÞÑ

text: p´

text: q

text: and

text: T

text: d

text: w

text: is

text: diagonal

text: and

text: satisﬁes

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text: symmetry

text: relation

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text: m

text: m,

text: m

text: so

text: it

text: suﬃces

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text: compute

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text: w

text: ΓdW

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text: l

text: d

text: l

text: l

text: There

text: is

text: a

text: coinc˜iden`ce

text: o˘f

text: form

text: among

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text: m`atrice˘s

text: in

text: the

text: functi¸onal

text: equations

text: of

text: the

text: Whittaker

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text: cons`tant

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text: Eisenstein`

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text: q

text: ΓdW

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text: π

text: ´2

text: Σd``

text: Σd`´

text: i

text: Σd´´

text: Σd´`

text: d

text: i

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text: u

text: Γ

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text: Γ

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text: So

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text: compute

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text: p

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text: q

text: ˆ

text: ˆ

text: d

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text: ΓdW

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text: d

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text: l

text: l

text: r

text: it

text: is

text: suﬃcient

text: to

text: compute

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text: has

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text: We

text: may

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text: symmetries

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text: F˘dp

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text: d,m

text: dm,d

text: d

text: ,m

text: m˘,d

text: p

text: q

text: For

text: the

text: moment,

text: we

text: assume

text: Re

text: u

text: is

text: large.

text: From

text: and

text: we

text: have

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text: Γ

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text: Γ

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text: d

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text: dπ

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text: When

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text: ˜ p `q ˘ ` ˘ ¸ HIGHER WEIGHT ON GL 3 , II: THE CUSP FORMS. 33 ` ˘ ` ˘ ` ˘ ` ˘ Therefore `` ` ˘ D p q p ` qD p q “ ´ Γ 1 u ` ` Γ 1 u ` d wl ΓdW u, 1 d wl π 12 Γ ´2u Σd`` Σd´` i Γ 1´u2 Σd´´ Σd`´ 2 2 ˆ D p q D p qR p´ q W p ´ qD p q D p q d w d w d i d 0, u d w d w , 2 3 3 2 ˘ “ p´ q ` “ p´ q P t u so if 1 1 d δ 1 η, δ, η 0, 1 , then ` ˘ d ` ˘ ` ˘ ÿ ˘D p q p ` qD p q (62) ud, d w ΓdW u,` 1 ˘ d w ` ˘ ` ˘ d l l ´ ` ` p q ´ ´ ` ´ “ p´ q iη d Γ 1 η u ˘ ´ 2d ! Γ d m u Γ d m u 1 d ´ ´2 c ud, i m p ` q p ´ q 2´ ` ´2 , 2d 1 Γ η u ď ” m m d m ! d m ! Γ d u Γ d 1 u 2 0 m δ 2 2 ` ˘ ` ˘ ˘ D p q p ``qD ˘p `q ˘ (63) ud,´ d w ΓdW u, 1 d w d 1 l l ` ´ ` ` d ´ ` ˘ ` ˘ “ ´p´ q ÿiη 1 d Γ 1 η u Γ 1 u 1 d ´ ´2 ´ `2 ` ˘ ` ˘ 2d 2 Γ η u Γ u 1 2 2 p ´ q ´ ´ ´ ´ ` ´ ˆ ˘ ´ 2d 1 ! Γ d 1 m u Γ d 1 m u c ud, i mm p ` q p ´ q 2´ ` ´ 2 , ď ” ´ m m d m ! d m ! Γ d u Γ d 1 u 0 m 1 δ 2 2 “ “ ‰ ˘ p´ q “ ˘p´ q where c 1, c 1, m 0. The sign parity ud, is determined by 1 m 1 d in 0 2 m´p´ q “ ˘p´ q m the ﬁrst case, and 1 m 1 d in the second. Similarly, (64) ` ˘ d ` ˘ ` ˘ ` ˘ ÿ ˘D p q p ` qD p q p ` q ud, d w ΓdW u`, 1 ˘ d w ΓdW u , 1 ` ˘ ` ˘ ` ˘ d l 1 l 2 ´ ` ` p q ´ ´ ` ´ ´ ` “ p´ q iη d Γ 1 η u1 ´ 2d ! Γ d m u1 Γ d m u1 Γ 1 m u2 1 d ´ ´2 c ud,εi m p ` q p ´ q ´2 ` ´2 ´ 2´ , 2d 1 Γ η u1 ď ” m m d m ! d m ! Γ d u1 Γ d 1 u1 Γ 1 m u2 2 0 m δ 2 2 2 (65) ` ˘ ` ˘ ˘ D p q p `` qD ˘p `q p ˘ ` q ud,´ d w ΓdW u , 1 d w ΓdW u , 1 d 1 l 1 l 2 ` ´ ` ` d ´ ` ˘ ` ˘ ` ˘ “ ´p´ q ÿiη 1 d Γ 1 η u1 Γ 1 u1 1 d 2d´2 Γ η´u21 Γ ´u12`1 ` ˘ ` ˘ ` ˘ 2 2 p ´ q ´ ´ ´ ´ ` ´ ´ ` ˆ ´ ´ 2d 1 ! Γ d 1 m u1 Γ d 1 m u1 Γ 1 m u2 c ud, εi mm p ` q p ´ q 2´ ` ´ 2 ´ 2´ , ď ” ´ m m d m ! d m ! Γ d u1 Γ d 1 u1 Γ 1 m u2 0 m 1 δ 2 2 2 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: p

text: `q

text: HIGHER

text: WEIGHT

text: ON

text: GL

text: II:

text: THE

text: CUSP

text: FORMS.

text: Therefore

text: D

text: p

text: q

text: p

text: qD

text: p

text: q

text: Γ

text: u

text: Γ

text: u

text: d

text: wl

text: ΓdW

text: u,

text: d

text: wl

text: π

text: Γ

text: Σd``

text: Σd´`

text: i

text: Γ

text: Σd´´

text: Σd`´

text: ˆ

text: D

text: p

text: q

text: D

text: p

text: qR

text: p´

text: q

text: W

text: p

text: qD

text: p

text: q

text: D

text: p

text: q

text: d

text: w

text: d

text: w

text: d

text: i

text: d

text: u

text: d

text: w

text: d

text: w

text: p´

text: q

text: p´

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text: P

text: t

text: u

text: so

text: if

text: d

text: δ

text: η,

text: δ,

text: η

text: then

text: d

text: ÿ

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text: qD

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text: l

text: p

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text: Γ

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text: m

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text: Γ

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text: d

text: ´2

text: c

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text: Γ

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text: ``qD

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text: d

text: w

text: ΓdW

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text: d

text: l

text: l

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text: ´p´

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text: ÿiη

text: d

text: Γ

text: η

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text: Γ

text: u

text: d

text: ´2

text: Γ

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text: Γ

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text: ˆ

text: Γ

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text: m

text: u

text: Γ

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text: ď

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text: m

text: d

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text: Γ

text: d

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text: Γ

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text: δ

text: p´

text: q

text: ˘p´

text: q

text: where

text: c

text: c

text: m

text: The

text: sign

text: parity

text: ud,

text: is

text: determined

text: by

text: m

text: d

text: in

text: m´p´

text: q

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text: m

text: the

text: ﬁrst

text: case,

text: and

text: m

text: d

text: in

text: the

text: second.

text: Similarly,

text: d

text: ÿ

text: ˘D

text: p

text: q

text: p

text: qD

text: p

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text: p

text: q

text: ud,

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text: ΓdW

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text: 40 JACK BUTTCANE [8] NIST Digital Library of Mathematical Functions. F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller and B. V. Saunders, eds. [9] W. Duke, J. B. Friedlander, and H. Iwaniec, The subconvexity problem for Artin L-functions, Invent. Math. 149 (2002), no. 3, 489–577. 1 [10] I. M. Gel fand, M. I. Graev, and I. I. Pyatetskii-Shapiro, Generalized functions. Vol. 6, AMS Chelsea Publishing, Providence, RI, 2016. Representation theory and automorphic functions, Translated from the 1966 Russian original by K. A. Hirsch, Reprint of the 1969 English translation. p q [11] D. Goldfeld, Automorphic forms and L-functions for the group GL n, R , Cambridge Studies in Ad- vanced Mathematics, vol. 99, Cambridge University Press, Cambridge, 2006. With an appendix by Kevin A. Broughan. [12] I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, Eighth, Elsevier/Academic Press, Amsterdam, 2015. Translated from the Russian, Translation edited and with a preface by Daniel Zwillinger and Victor Moll, Revised from the seventh edition. [13] Harish-Chandra, Automorphic forms on semisimple Lie groups, Notes by J. G. M. Mars. Lecture Notes in Mathematics, No. 62, Springer-Verlag, Berlin-New York, 1968. [14] Roger Howe, K-type structure in the principal series of GL . I, Analysis on homogeneous spaces and 3 representation theory of Lie groups, Okayama–Kyoto (1997), 2000, pp. 77–98. [15] Henryk Iwaniec, Introduction to the spectral theory of automorphic forms, Biblioteca de la Revista Matem´atica Iberoamericana. [Library of the Revista Matem´atica Iberoamericana], Revista Matem´atica Iberoamericana, Madrid, 1995. [16] H. Jacquet and J. A. Shalika, On Euler products and the classiﬁcation of automorphic forms. II, Amer. J. Math. 103 (1981), no. 4, 777–815. [17] Henry H. Kim, Functoriality for the exterior square of GL and the symmetric fourth of GL , J. Amer. 4 2 Math. Soc. 16 (2003), no. 1, 139–183 (electronic). With appendix 1 by Dinakar Ramakrishnan and appendix 2 by Kim and Peter Sarnak. [18] Anthony W. Knapp, Representation theory of semisimple groups, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 2001. An overview based on examples, Reprint of the 1986 original. [19] R. P. Langlands, On the functional equations satisﬁed by Eisenstein series, Lecture Notes in Mathemat- ics, Vol. 544, Springer-Verlag, Berlin-New York, 1976. [20] Hans Maass, Die Diﬀerentialgleichungen in der Theorie der elliptischen Modulfunktionen, Math. Ann. 125 (1952), 235–263 (1953). p q [21] H. Manabe, T. Ishii, and T. Oda, Principal series Whittaker functions on SL 3, R , Japan. J. Math. (N.S.) 30 (2004), no. 1, 183–226. [22] Stephen D. Miller and Wilfried Schmid, Adelization of automorphic distributions and mirabolic Eisen- stein series, Representation theory and mathematical physics, 2011, pp. 289–334. p q p q [23] Tadashi Miyazaki, The structures of standard g, K -modules of SL 3, R , Glas. Mat. Ser. III 43(63) (2008), no. 2, 337–362. p q [24] , Whittaker functions for generalized principal series representations of SL 3, R , New viewpoints of representation theory and noncommutative harmonic analysis, 2010, pp. 103–110. [25] M. Scott Osborne and Garth Warner, The theory of Eisenstein systems, Pure and Applied Mathematics, vol. 99, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. [26] A. Selberg, Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series, J. Indian Math. Soc. (N.S.) 20 (1956), 47–87. [27] J. A. Shalika, The multiplicity one theorem for GL , Ann. of Math. (2) 100 (1974), 171–193. n p q [28] I J Vahutinski˘i, Irreducible unitary representations of the group GL 3, R of real matrices of the third order, Mathematics of the USSR-Sbornik 4 (1968), no. 2, 273. Mathematics Department, 244 Mathematics Building, Buffalo, NY 14260, USA Email address: buttcane@buffalo.edu ##LTLine##

text: JACK

text: BUTTCANE

label: reference

text: NIST

text: Digital

text: Library

text: of

text: Mathematical

text: Functions.

text: F.

text: W.

text: J.

text: Olver,

text: A.

text: B.

text: Olde

text: Daalhuis,

text: D.

text: W.

text: Lozier,

text: B.

text: I.

text: Schneider,

text: R.

text: F.

text: Boisvert,

text: C.

text: W.

text: Clark,

text: B.

text: R.

text: Miller

text: and

text: B.

text: V.

text: Saunders,

text: eds.

text: W.

text: Duke,

text: J.

text: B.

text: Friedlander,

text: and

text: H.

text: Iwaniec,

text: The

text: subconvexity

text: problem

text: for

text: Artin

text: L-functions,

text: Invent.

text: Math.

text: no.

text: I.

text: M.

text: Gel

text: fand,

text: M.

text: I.

text: Graev,

text: and

text: I.

text: I.

text: Pyatetskii-Shapiro,

text: Generalized

text: functions.

text: Vol.

text: AMS

text: Chelsea

text: Publishing,

text: Providence,

text: RI,

text: Representation

text: theory

text: and

text: automorphic

text: functions,

text: Translated

text: from

text: the

text: Russian

text: original

text: by

text: K.

text: A.

text: Hirsch,

text: Reprint

text: of

text: the

text: English

text: translation.

text: p

text: q

text: D.

text: Goldfeld,

text: Automorphic

text: forms

text: and

text: L-functions

text: for

text: the

text: group

text: GL

text: n,

text: R

text: Cambridge

text: Studies

text: in

text: Ad-

text: vanced

text: Mathematics,

text: vol.

text: Cambridge

text: University

text: Press,

text: Cambridge,

text: With

text: an

text: appendix

text: by

text: Kevin

text: A.

text: Broughan.

text: I.

text: S.

text: Gradshteyn

text: and

text: I.

text: M.

text: Ryzhik,

text: Table

text: of

text: integrals,

text: series,

text: and

text: products,

text: Eighth,

text: Elsevier/Academic

text: Press,

text: Amsterdam,

text: Translated

text: from

text: the

text: Russian,

text: Translation

text: edited

text: and

text: with

text: a

text: preface

text: by

text: Daniel

text: Zwillinger

text: and

text: Victor

text: Moll,

text: Revised

text: from

text: the

text: seventh

text: edition.

text: Harish-Chandra,

text: Automorphic

text: forms

text: on

text: semisimple

text: Lie

text: groups,

text: Notes

text: by

text: J.

text: G.

text: M.

text: Mars.

text: Lecture

text: Notes

text: in

text: Mathematics,

text: No.

text: Springer-Verlag,

text: Berlin-New

text: York,

text: Roger

text: Howe,

text: K-type

text: structure

text: in

text: the

text: principal

text: series

text: of

text: GL

text: I,

text: Analysis

text: on

text: homogeneous

text: spaces

text: and

text: representation

text: theory

text: of

text: Lie

text: groups,

text: Okayama–Kyoto

text: pp.

text: Henryk

text: Iwaniec,

text: Introduction

text: to

text: the

text: spectral

text: theory

text: of

text: automorphic

text: forms,

text: Biblioteca

text: de

text: la

text: Revista

text: Matem´atica

text: Iberoamericana.

text: [Library

text: of

text: the

text: Revista

text: Matem´atica

text: Iberoamericana],

text: Revista

text: Matem´atica

text: Iberoamericana,

text: Madrid,

text: H.

text: Jacquet

text: and

text: J.

text: A.

text: Shalika,

text: On

text: Euler

text: products

text: and

text: the

text: classiﬁcation

text: of

text: automorphic

text: forms.

text: II,

text: Amer.

text: J.

text: Math.

text: no.

text: Henry

text: H.

text: Kim,

text: Functoriality

text: for

text: the

text: exterior

text: square

text: of

text: GL

text: and

text: the

text: symmetric

text: fourth

text: of

text: GL

text: J.

text: Amer.

text: Math.

text: Soc.

text: no.

text: (electronic).

text: With

text: appendix

text: by

text: Dinakar

text: Ramakrishnan

text: and

text: appendix

text: by

text: Kim

text: and

text: Peter

text: Sarnak.

text: Anthony

text: W.

text: Knapp,

text: Representation

text: theory

text: of

text: semisimple

text: groups,

text: Princeton

text: Landmarks

text: in

text: Mathematics,

text: Princeton

text: University

text: Press,

text: Princeton,

text: NJ,

text: An

text: overview

text: based

text: on

text: examples,

text: Reprint

text: of

text: the

text: original.

text: R.

text: P.

text: Langlands,

text: On

text: the

text: functional

text: equations

text: satisﬁed

text: by

text: Eisenstein

text: series,

text: Lecture

text: Notes

text: in

text: Mathemat-

text: ics,

text: Vol.

text: Springer-Verlag,

text: Berlin-New

text: York,

text: Hans

text: Maass,

text: Die

text: Diﬀerentialgleichungen

text: in

text: der

text: Theorie

text: der

text: elliptischen

text: Modulfunktionen,

text: Math.

text: Ann.

text: p

text: q

text: H.

text: Manabe,

text: T.

text: Ishii,

text: and

text: T.

text: Oda,

text: Principal

text: series

text: Whittaker

text: functions

text: on

text: SL

text: R

text: Japan.

text: J.

text: Math.

text: (N.S.)

text: no.

text: Stephen

text: D.

text: Miller

text: and

text: Wilfried

text: Schmid,

text: Adelization

text: of

text: automorphic

text: distributions

text: and

text: mirabolic

text: Eisen-

text: stein

text: series,

text: Representation

text: theory

text: and

text: mathematical

text: physics,

text: pp.

text: p

text: q

text: p

text: q

text: Tadashi

text: Miyazaki,

text: The

text: structures

text: of

text: standard

text: g,

text: K

text: -modules

text: of

text: SL

text: R

text: Glas.

text: Mat.

text: Ser.

text: III

text: no.

text: p

text: q

text: Whittaker

text: functions

text: for

text: generalized

text: principal

text: series

text: representations

text: of

text: SL

text: R

text: New

text: viewpoints

text: of

text: representation

text: theory

text: and

text: noncommutative

text: harmonic

text: analysis,

text: pp.

text: M.

text: Scott

text: Osborne

text: and

text: Garth

text: Warner,

text: The

text: theory

text: of

text: Eisenstein

text: systems,

text: Pure

text: and

text: Applied

text: Mathematics,

text: vol.

text: Academic

text: Press,

text: Inc.

text: [Harcourt

text: Brace

text: Jovanovich,

text: Publishers],

text: New

text: York-London,

text: A.

text: Selberg,

text: Harmonic

text: analysis

text: and

text: discontinuous

text: groups

text: in

text: weakly

text: symmetric

text: Riemannian

text: spaces

text: with

text: applications

text: to

text: Dirichlet

text: series,

text: J.

text: Indian

text: Math.

text: Soc.

text: (N.S.)

text: J.

text: A.

text: Shalika,

text: The

text: multiplicity

text: one

text: theorem

text: for

text: GL

text: Ann.

text: of

text: Math.

text: n

text: p

text: q

text: I

text: J

text: Vahutinski˘i,

text: Irreducible

text: unitary

text: representations

text: of

text: the

text: group

text: GL

text: R

text: of

text: real

text: matrices

text: of

text: the

text: third

text: order,

text: Mathematics

text: of

text: the

text: USSR-Sbornik

text: no.

text: Mathematics

text: Department,

text: Mathematics

text: Building,

text: Buffalo,

text: NY

text: USA

text: Email

text: address:

text: buttcane@buffalo.edu

text: ##LTLine##


text: p q HIGHER WEIGHT ON GL 3 , II: THE CUSP FORMS. 9 p q{ p q R R On functions of P SL 3, SO 3, these become [11, eqs (6.1.1)] ˝ “ ´ B ´ B ` B B ´ p ` qB ´ B ´ B ´ B B ∆ y2 2 y2 2 y y y2 x2 y2 2 y2 2 y2 2 2y2x , ˝ “1 ´ 1By1B `2 y2 B 1B2 y´1 y2 B 1 B 2 ` 2 Bx3 B 1´x1 2 xB2 B B1 2 x1 x3 ∆ y2y 2 y y2 2 y3y 2 y y2 2 2y2y x 2 ` p´1 2 y`1 y2q 1B2 By1 y´2 1 B2 xB3 y`1 1 2B x2B y1B ` 1 2 2 x1B x3B y2 x2 y2 y2y 2 y2y 2 2y2y2 2y2y2x 2 ` B 2 ´ 2B 1 `2 x3 y2 B B1 2`x1p y2` q1 2B x1`x2 xB3 ´ 1B2 2 x2 x3 y2 2 y2 2 2y2x x2 y2 y2 2 y2 2 y2 2 . 1 y1 2 y2 1 2 x1 x3 2 2 1 x3 1 x1 2 x2 We will require expressions for the full operators. The Lie algebra acts on smooth functions on K on the left by the operators [3, Section 3.8] J “ p B ` B ´ B q i cos α cot β sin α cos α csc β , 1 α β γ J “ p Ba´ B ´ B q i sin α cot β cos α sin α csc β , 2 α β γ with p´J ˘ J q D “ p ` q ´ 1p 1 ˘ q D i d 1 d d 1 m m 1 d 1˘ . 1 2 m ,m m 1,m We deﬁne the operators T “ p B ` B q T “T T ` T p ´ B q y x , 1 y , T1 “ 1 B2 x3 x1 T4 “T3p 2 ` 1B ´ 2 By2 q ` T T y y , 1 y y , T2 “ 1B2 x3 T5 “T2T ´ T2 `y2 B 1 Ty1 1 3 y , y , 3 2 x2 6 1 2 3 1 y1 3 p q then as functions of x , x , x , y , y , α, β, γ, the full Casimir operators on SL 3 are given by 1 2 3 1 2 “ ˝ ` J T ´ J T ` B T (13) ∆ ∆ i i , 1 “ 1˝ ` J1pT1 ` B 2T2q ´ αJ 3T ` B T (14) ∆ ∆ i i . 2 2 1 4 α 2 2 5 α 6 r r p D q ÿ Let C G, d be the space of smooth functions on G that lie in the span of the entries of D P p D q d. That is f C G, d if and only if p q “ p qD p q r f xyk f 1 xy d 1 k . m ,m m ,m | 1| | |ď m , m d P p D q ÿ ż D ÿ For f C G, d , taking the expansion into Wigner -functions rand evaluating at the identity gives p q “ p ` q p 1qD p 1q 1 “ P p q f xyk 2d 1 f xykkż d k dk d f xyk , m,m m | |ď | |ď r K m d m d where pP qp q “ p ` q p qD p q d f g 2d 1 f gk d k dk m m,m K D is the projection onto the span of the m-th column of d. ##LTLine## ##LTLine## ##LTLine##

text: p

text: q

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text: WEIGHT

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text: THE

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text: J

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text: B

text: B

text: B

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text: cos

text: α

text: cot

text: β

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text: α

text: β

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text: the

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text: SL

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text: ##LTLine##

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text: (ii) using diﬀerent dust opacities due to dust-grain growth (e.g., ??). We varied the abundance of wa- 1.2 ter vapor in the disk atmosphere to simulate the 17.75um 24.00um eﬀects of (iii) increasing/decreasing the strength 61.32um 1 94.17um of UV photoreactions. We found that in the cases 482.99um 682.66um for which the H O snowline is more distant from 0.8 933.28um 2 the central star, and the fractional abundance of 0.6 H2O gas in the disk atmosphere is lower than that in the originally adopted T Tauri disk, the line 0.4 ﬂuxes from the hot inner disk midplane inside the H O snowline are more dominant from those from 0.2 2 the outer disk (e.g., ?). 0 1 10 An analysis of spectral energy distributions (SEDs) r [AU] classiﬁed Herbig Ae/Be stars into two groups 1.2 17.75um (group I/II, ?). Group I sources show both power- 24.00um 61.32um law and blackbody components up to far-infrared 1 94.17um 482.99um wavelengths in their SEDs. SEDs of group II 682.66um 0.8 933.28um sources can be well modeled with only a single power law from mid- to far-infrared wavelengths. 0.6 ? suggested that group I sources have a ﬂaring disk while the group II disks are geometrically 0.4 ﬂat. ? proposed a possible evolutionary scenario 0.2 where a group I ﬂaring disk evolves into a group II ﬂat disk through grain growth and settling of 0 1 10 100 grains onto the disk midplane (see also ??). r [AU] However, recent high-spatial resolution obser- vations at various wavelengths have revealed a more complex structure in disks, with inner holes and/or gaps toward many group I sources such as HD100546 (e.g., ???), HD142527 (e.g., ???), and Fig. 12.— The radial distributions of the nor- HD169142 (e.g., ??). On the other hand, there malized cumulative ﬂux for seven pure rotational is little evidence for inner holes and/or gaps re- ortho-H O lines at λ=17.75µm (black long dashed 2 ported toward group II disks, and they seem to line), 24.00µm (blue dashed line), 61.32µm (vi- have a radially continuous structure (e.g., ?). ? olet dotted line), 94.17µm (green dashed dotted and ? suggested that most group I sources can be line), 482.99µm (brown dashed two dotted line), classiﬁed as (pre-)transitional disks. Transitional 682.66µm (red solid line), and 933.28µm (orange and pre-transitional disks are protoplanetary disks long dashed dotted line). These are candidate with an inner hole and/or gaps indicated by a lack ortho-H O lines to trace the emission from the hot 2 of near-infrared/mid-infrared excess in their SEDs water vapor within the H O snowline. The verti- 2 (e.g., ??). They pointed out that there is no sig- cal straight lines display the positions of r=8 au niﬁcant diﬀerence in age between groups I and II (orange dotted line) and 14 au (grey dotted line), sources (??). Therefore, recent studies (e.g., ??) respectively. We normalized the cumulative ﬂux suggested that both group I and II sources had ex- of each line using the values at r = 30 au (top perienced diﬀerent evolutionary paths from some panel) and at r = 300 au (bottom panel). We as- common primordial and continuous ﬂaring disks. sume that the inclination angle of the disk i is 0 Here we note that ?, ?, and ? suggested that some degree in making these ﬁgures. geometrically ﬂat disks (group II disk) have gaps. ? suggested ﬂat disks with gaps are most likely 17 xulF evitalumuC dezilamroN xulF evitalumuC dezilamroN ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: (ii)

text: using

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text: vapor

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text: disk

text: atmosphere

text: to

text: simulate

text: the

text: eﬀects

text: of

text: (iii)

text: increasing/decreasing

text: the

text: strength

text: of

text: UV

text: photoreactions.

text: We

text: found

text: that

text: in

text: the

text: cases

text: for

text: which

text: the

text: H

text: O

text: snowline

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text: more

text: distant

text: from

text: the

text: central

text: star,

text: and

text: the

text: fractional

text: abundance

text: of

text: gas

text: in

text: the

text: disk

text: atmosphere

text: is

text: lower

text: than

text: that

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text: the

text: originally

text: adopted

text: T

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text: disk,

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text: outer

text: disk

text: (e.g.,

text: An

text: analysis

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text: spectral

text: energy

text: distributions

text: (SEDs)

text: r

text: [AU]

text: classiﬁed

text: Herbig

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text: into

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text: power-

text: law

text: and

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text: components

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text: wavelengths

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text: their

text: SEDs.

text: SEDs

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text: group

text: II

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text: well

text: modeled

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text: only

text: a

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text: law

text: from

text: mid-

text: to

text: far-infrared

text: wavelengths.

text: suggested

text: that

text: group

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text: have

text: a

text: ﬂaring

text: disk

text: while

text: the

text: group

text: II

text: disks

text: are

text: geometrically

text: ﬂat.

text: proposed

text: a

text: possible

text: evolutionary

text: scenario

text: where

text: a

text: group

text: I

text: ﬂaring

text: disk

text: evolves

text: into

text: a

text: group

text: II

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text: disk

text: through

text: grain

text: growth

text: and

text: settling

text: of

text: grains

text: onto

text: the

text: disk

text: midplane

text: (see

text: also

text: r

text: [AU]

text: However,

text: recent

text: high-spatial

text: resolution

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text: vations

text: at

text: various

text: wavelengths

text: have

text: revealed

text: a

text: more

text: complex

text: structure

text: in

text: disks,

text: with

text: inner

text: holes

text: and/or

text: gaps

text: toward

text: many

text: group

text: I

text: sources

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text: as

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text: (e.g.,

text: and

text: Fig.

text: The

text: radial

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text: nor-

text: (e.g.,

text: On

text: the

text: other

text: hand,

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text: malized

text: cumulative

text: ﬂux

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text: rotational

text: is

text: little

text: evidence

text: for

text: inner

text: holes

text: and/or

text: gaps

text: re-

text: ortho-H

text: O

text: lines

text: at

text: (black

text: long

text: dashed

text: ported

text: toward

text: group

text: II

text: disks,

text: and

text: they

text: seem

text: to

text: line),

text: (blue

text: dashed

text: line),

text: (vi-

text: have

text: a

text: radially

text: continuous

text: structure

text: (e.g.,

text: olet

text: dotted

text: line),

text: (green

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text: and

text: suggested

text: that

text: most

text: group

text: I

text: sources

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text: be

text: line),

text: (brown

text: dashed

text: two

text: dotted

text: line),

text: classiﬁed

text: as

text: (pre-)transitional

text: disks.

text: Transitional

text: (red

text: solid

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text: and

text: (orange

text: and

text: pre-transitional

text: disks

text: are

text: protoplanetary

text: disks

text: long

text: dashed

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text: line).

text: These

text: are

text: candidate

text: with

text: an

text: inner

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text: O

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text: from

text: the

text: hot

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text: excess

text: in

text: their

text: SEDs

text: water

text: vapor

text: within

text: the

text: H

text: O

text: snowline.

text: The

text: verti-

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text: pointed

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text: there

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text: of

text: au

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text: diﬀerence

text: in

text: age

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text: groups

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text: and

text: II

text: (orange

text: dotted

text: line)

text: and

text: au

text: (grey

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text: line),

text: sources

text: Therefore,

text: recent

text: studies

text: (e.g.,

text: respectively.

text: We

text: normalized

text: the

text: cumulative

text: ﬂux

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text: that

text: both

text: group

text: I

text: and

text: II

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text: had

text: ex-

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text: the

text: values

text: at

text: r

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text: perienced

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text: panel)

text: and

text: at

text: r

text: au

text: (bottom

text: panel).

text: We

text: as-

text: common

text: primordial

text: and

text: continuous

text: ﬂaring

text: disks.

text: sume

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text: the

text: inclination

text: angle

text: of

text: the

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text: is

text: Here

text: we

text: note

text: that

text: and

text: suggested

text: that

text: some

text: degree

text: in

text: making

text: these

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text: geometrically

text: ﬂat

text: disks

text: (group

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text: have

text: gaps.

text: suggested

text: ﬂat

text: disks

text: with

text: gaps

text: are

text: most

text: likely

text: xulF

text: evitalumuC

text: dezilamroN

text: xulF

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text: dezilamroN

text: ##LTFigure##

text: ##LTFigure##

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text: 1.2 10000 1.2 10000 1 1 0.8 0.8 0.6 1000 0.6 1000 0.4 17.75T µgams 0.4 682.66T µgams 24.00 µm 63.32 µm 61.32 µm 538.29 µm 0.2 94.17 µm 0.2 12.40 µm 482.99 µm 12.45 µm 682.66 µm 4.96 µm 0 933.28 µm 100 0 4.43 µm 100 -0.4 -0.2 0 0.2 0.4 -0.4 -0.2 0 0.2 0.4 z/r z/r 1.2 10000 1.2 10000 1 1 0.8 0.8 0.6 1000 0.6 1000 0.4 17.75T µgams 0.4 682.66T µgams 24.00 µm 63.32 µm 61.32 µm 538.29 µm 0.2 94.17 µm 0.2 12.40 µm 482.99 µm 12.45 µm 682.66 µm 4.96 µm 0 933.28 µm 100 0 4.43 µm 100 -0.4 -0.2 0 0.2 0.4 -0.4 -0.2 0 0.2 0.4 z/r z/r 1.2 10000 1.2 10000 1 1 0.8 1000 0.8 1000 0.6 0.6 0.4 17.75T µgams 100 0.4 682.66T µgams 100 24.00 µm 63.32 µm 61.32 µm 538.29 µm 0.2 94.17 µm 0.2 12.40 µm 482.99 µm 12.45 µm 682.66 µm 4.96 µm 0 933.28 µm 10 0 4.43 µm 10 -0.4 -0.2 0 0.2 0.4 -0.4 -0.2 0 0.2 0.4 z/r z/r Fig. 13.— The vertical distributions of the normalized cumulative line emissivity at r =5 au (top two panels), r =10 au (middle two panels), and r =30 au (bottom two panels), and of the gas temperature T at Kelvin g (gray dotted line). The left three panels show the distributions for seven pure rotational ortho-H O lines at 2 λ=17.75µm (red solid line), 24.00µm (black long dashed line), 61.32µm (blue dashed line), 94.17µm (brown dashed two dotted line), 482.99µm (green dashed dotted line), 682.66µm (violet long dashed dotted line), and 933.28µm (orange dotted line. The right three panels show the distributions for seven pure rotational ortho- H O lines at λ=682.66µm (violet long dashed dotted line), 63.32µm (red solid line), 538.29µm (black long 2 dashed line), 12.40µm (blue dashed line), 12.45µm (brown dashed two dotted line), 4.96µm (green dashed dotted line), and 4.43µm (orange dotted line). We normalized the cumulative emissivity of each line using −∞ the values at z = . We assume that the inclination angle of the disk i is 0 degree in making these ﬁgures. 24 ytivissimE evitalumuC dezilamroN ytivissimE evitalumuC dezilamroN ytivissimE evitalumuC dezilamroN ]K[ erutarepmeT saG ]K[ erutarepmeT saG ]K[ erutarepmeT saG ytivissimE evitalumuC dezilamroN ytivissimE evitalumuC dezilamroN ytivissimE evitalumuC dezilamroN ]K[ erutarepmeT saG ]K[ erutarepmeT saG ]K[ erutarepmeT saG ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: µgams

text: µgams

text: µm

text: µm

text: µm

text: µm

text: µm

text: µm

text: µm

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text: µgams

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text: z/r

text: z/r

text: Fig.

text: The

text: vertical

text: distributions

text: of

text: the

text: normalized

text: cumulative

text: line

text: emissivity

text: at

text: r

text: au

text: (top

text: two

text: panels),

text: r

text: au

text: (middle

text: two

text: panels),

text: and

text: r

text: au

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text: panels),

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text: The

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text: ytivissimE

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text: Table B.1 − Candidate ortho-H O line parameters and total line fluxes (11 1000µm) 2 J λ1 Freq. A E HAe ﬂux2,3 TT ﬂux3,4 KaKc −ul up − − [µm] [GHz] [s 1] [K] [W m 2] [W m 2] × − × − × − 7 -6 11.878 25239.615 6.653 10 3 2006.8 3.4 10 19 1.8 10 22 70 25 × − × − × − 7 -6 13.005 23052.310 2.585 10 4 1749.8 2.9 10 18 1.1 10 21 61 16 × − × − × − 6 -5 15.489 19354.618 4.138 10 4 1503.6 6.9 10 18 2.6 10 21 61 14 × − × − × − 7 -7 16.329 18359.435 7.756 10 5 2006.8 3.8 10 19 1.6 10 22 70 25 × − × − × − 6 -6 16.728 17921.765 2.153 10 5 1503.6 3.2 10 19 1.1 10 22 61 16 × − × − × − 6 -5 17.754 16885.840 2.909 10 3 1278.5 4.1 10 17 2.3 10 20 52 05 × − × − × − 7 -7 19.532 15348.791 1.269 10 3 1749.8 8.3 10 18 3.5 10 21 61 16 × − × − × − 7 -8 19.638 15265.853 1.900 10 6 2006.8 7.6 10 21 3.1 10 24 70 27 × − × − × − 7 -7 21.115 14197.932 2.078 10 3 1524.8 2.0 10 17 9.9 10 21 52 07 × − × − × − 7 -8 21.185 14151.161 6.154 10 6 1749.8 4.8 10 20 1.7 10 23 61 18 × − × − × − 7 -7 21.571 13897.693 3.474 10 3 2006.8 1.1 10 17 5.5 10 21 70 43 × − × − × − 5 -5 23.996 12493.205 2.696 10 4 1067.7 9.4 10 18 6.4 10 21 50 05 × − × − × − 6 -6 25.247 11874.160 3.803 10 3 1503.6 2.6 10 17 1.9 10 20 61 34 × − × − × − 6 -7 29.339 10218.246 1.859 10 5 1503.6 2.5 10 19 1.3 10 22 61 16 × − × − × − 8 -9 29.851 10043.001 1.605 10 4 1805.9 1.2 10 18 4.7 10 22 54 09 × − × − × − 6 -7 33.074 9064.381 5.942 10 5 1278.5 1.4 10 18 1.3 10 21 52 07 × − × − × − 5 -5 33.833 8860.835 4.552 10 3 1067.7 3.2 10 17 7.1 10 20 50 23 × − × − × − 7 -8 36.750 8157.523 1.123 10 4 2006.8 4.3 10 19 1.6 10 22 70 45 × − × − × − 4 -4 37.984 7892.616 2.629 10 3 702.3 3.1 10 17 1.0 10 19 41 14 × − × − × − 9 -10 40.702 7365.464 4.815 10 4 1957.1 2.1 10 18 1.1 10 21 45 110 × − × − × − 7 -8 47.601 6297.997 4.268 10 4 1749.8 2.3 10 18 2.2 10 21 61 36 × − × − × − 6 -7 49.334 6076.834 1.195 10 4 1503.6 1.3 10 18 1.8 10 21 61 34 × − × − × − 5 -6 52.864 5671.021 1.442 10 4 1067.7 3.7 10 18 1.4 10 20 50 25 × − × − × − 7 -8 53.455 5608.351 5.030 10 4 1339.8 5.2 10 18 1.6 10 20 43 18 × − × − × − 7 -8 57.394 5223.441 7.203 10 4 1524.8 4.1 10 18 1.1 10 20 52 27 × − × − × − 5 -6 61.316 4889.280 2.686 10 4 878.1 5.9 10 18 3.5 10 20 41 16 × − × − × − 6 -7 94.172 3183.464 3.387 10 4 1278.5 1.8 10 18 1.6 10 20 52 25 × − × − × − 4 -5 112.803 2657.666 1.670 10 4 702.3 1.8 10 18 2.3 10 20 41 14 × − × − × − 9 -10 114.454 2619.334 6.768 10 4 1729.3 1.0 10 18 8.1 10 21 27 110 × − × − × − 8 -9 116.350 2576.644 7.255 10 4 1447.5 1.4 10 18 1.5 10 20 36 09 × − × − × − 8 -7 159.051 1884.888 7.403 10 3 1615.3 1.1 10 18 1.7 10 20 45 52 × − × − × − 6 -7 159.400 1880.753 3.911 10 4 933.7 9.0 10 19 1.5 10 20 34 07 × − × − × − 8 -9 187.810 1595.252 2.947 10 4 1805.9 2.5 10 19 3.8 10 21 54 27 × − × − × − 6 -7 190.437 1574.232 2.927 10 4 1088.7 4.9 10 19 8.9 10 21 43 16 × − × − × − 6 -5 226.761 1322.065 2.334 10 3 795.5 6.0 10 19 1.2 10 20 25 32 × − × − × − 8 -9 229.206 1307.963 2.599 10 4 1615.3 2.0 10 19 3.9 10 21 45 18 × − × − × − 8 -7 231.248 1296.411 1.074 10 3 1274.1 3.7 10 19 6.8 10 21 27 34 × − × − × − 7 -6 234.531 1278.266 1.552 10 3 1399.8 3.6 10 19 6.7 10 21 43 52 × − × − × − 8 -7 256.593 1168.358 9.459 10 4 1805.9 1.8 10 19 3.5 10 21 54 61 × − × − × − 6 -5 258.816 1158.324 1.417 10 3 933.7 3.5 10 19 7.0 10 21 34 41 25 ##LTLine## ##LTLine## ##LTLine##

text: Table

text: Candidate

text: ortho-H

text: O

text: line

text: parameters

text: and

text: total

text: line

text: fluxes

text: J

text: λ1

text: Freq.

text: A

text: E

text: HAe

text: TT

text: KaKc

text: −ul

text: up

text: [µm]

text: [GHz]

text: [s

text: [K]

text: [W

text: m

text: [W

text: m

text: ##LTLine##

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text: ##LTLine##


text: C. The proﬁles of candidate mid-infrared H O lines to trace the H O snowline 2 2 Figure ?? shows the proﬁles of mid-infrared candidate H O lines which trace the hot water vapor within the ∼ − 2 H O snowline (λ 11 25µm). All of the lines in this Figure are also listed in Table ??.1. We discuss the 2 properties of these lines in Sections 3.2.3 and 4.4. Fig. 14.— The velocity proﬁles of mid-infrared ortho-H O lines at λ=11.88µm (top left), 13.01µm (top 2 right), 15.49µm (middle left), 16.33µm (middle right), 16.73µm (bottom left), and 17.75µm (bottom right), from the Herbig Ae disk. These exist between 11-25µm and are also the best mid-infrared candidate ortho- H O lines to trace the hot water vapor within the H O snowline. The parameters and total ﬂuxes of these 2 2 ortho-H O lines are reported in Table ??.1. Red solid lines are the emission line proﬁles from inside 8 2 ∼ au (=the inner high temperature region), blue dashed lines are those from inside 14 au ( inside the H O ∼ 2 snowline), green dotted lines are those from 14-30 au ( outside the H O snowline), and black dashed dotted 2 lines are those from the total area inside 30au. Fig. 14.— (Continued.) The velocity proﬁles of mid-infrared ortho-H O lines at λ=19.53µm (top left), 2 19.64µm (top right), 21.12µm (middle left), 21.19µm (middle right), 21.57µm (bottom left), and 24.00µm (bottom right), from the Herbig Ae disk. 27

text: C.

label: section

text: The

text: proﬁles

text: of

text: candidate

text: mid-infrared

text: H

text: O

text: lines

text: to

text: trace

text: the

text: H

text: O

text: snowline

text: Figure

text: shows

text: the

text: proﬁles

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text: candidate

text: H

text: O

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text: the

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text: lines

text: in

text: Sections

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text: Fig.

text: The

label: caption

text: velocity

text: proﬁles

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