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GENERIC REPRESENTATIONS FOR QUASI-SPLIT SIMILITUDEGROUPS
The 2020 Paul J. Sally, Jr. Midwest Representation Theory Conference
Chris Jantzen, East Carolina University
October 17, 2020
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2 GENERIC REPRESENTATIONS FOR QUASI-SPLIT SIMILITUDE GROUPS
1. Introduction
The following is drawn from The generic dual of p-adic groups and localLanglands parameters1, joint with Baiying Liu.
Let F be a p-adic field of characteristic 0.We focus on classical and similitude groups whose standard Levi factors
have the formM = GLn1 × · · · ×GLnk ×Gn0,
where Gn0 is a lower rank group of the same type, and whose Weyl groupacts as permutations and sign changes.
1Legal disclaimer: paper in preparation; corrections possible.
GENERIC REPRESENTATIONS FOR QUASI-SPLIT SIMILITUDE GROUPS 3
1.1. Groups considered. For quasi-split groups, fix a quadratic extensionE = F (
√ε). We let Gn be one of the following groups:
• classical groups: Gn is SO2n+1, Sp2n, SO2n, U2n+1, U2n, or SO∗2n+2
• similitude groups: Gn is GSp2n, GSO2n, GSpin2n+1, GSpin2n, GU2n+1,GU2n, GSO
∗2n+2, or GSpin∗2n+2
Parabolic subgroups and basic structure of these groups may be found in[Tad1], [Gol], [Asg], [H-S], [Xu] among other places.
4 GENERIC REPRESENTATIONS FOR QUASI-SPLIT SIMILITUDE GROUPS
1.2. Goals.
• Classify generic representations (with an eye toward applications as in[J-S], [Liu], [J-L])• Fill in the gaps in the tools needed.• Uniformize arguments across the different groups considered (as in
[Tad3], [M-T], [ACS])
GENERIC REPRESENTATIONS FOR QUASI-SPLIT SIMILITUDE GROUPS 5
2. Generic representations
2.1. Generic essentially discrete series. Generic discrete series were clas-sified for Gn = SO2n+1 and Sp2n in [Mui] and for SO2n in [J-L].
Notation.
• ν = | · | composed with determinant or similitude character• ×,o denote (normalized) parabolic induction• π essentially tempered if ∃ ε(π) ∈ R such that ν−ε(π)(π) is tempered
• β =
12ε if Gn = GSpin2n+1, GSpin2n with n = 0,ε if Gn = GSpin∗2n+2 or Gn = GSpin2n+1, GSpin2n with n > 0,0 if Gn is not a general spin group.
• c=usual outer automorphism–for SO2n, GSO2n, GSpin2n acts nontriv-ially on the simple roots; for SO∗2n+2, GSO
∗2n+2, GSpin
∗2n+2, acts triv-
ially on the F -data but nontrivially on the maximal quasi-split torus
6 GENERIC REPRESENTATIONS FOR QUASI-SPLIT SIMILITUDE GROUPS
We recall the following combination of results from [Zel] and [Jac]:
Theorem (general linear groups). Let τ be an irreducible unitary supercus-pidal representation of a general linear group. Let
[νmτ, νnτ ] = {νmτ, νm+1τ, . . . , νnτ}(with m − n ∈ Z). We define δ([νmτ, νnτ ]) to be the (unique) irreduciblequotient of νmτ×νm+1τ×· · ·×νnτ . Then δ([νmτ, νnτ ]) is essentially square-integrable, and every irreducible essentially square-integrable representationof a general linear groups has this form. Further, every such representationis generic.
GENERIC REPRESENTATIONS FOR QUASI-SPLIT SIMILITUDE GROUPS 7
Theorem (generic essentially discrete series). Let ∆i = [ν−aiτi, νbiτi], 1 ≤
i ≤ k, where τi is an irreducible unitary supercuspidal representation of ageneral linear group. Assume that if i < j has τi ∼= τj, then ai < bi < aj < bj.Let σ(e0) be an irreducible supercuspidal generic representation of Gn′(F ) andassume that for each i, one of the following holds (necessarily exclusive):
(1) ν1+βτi o σ(e0) is reducible, in which case ai ∈ β + (Z \ {0}) and ai ≥β − 1;
(2) ν12+βτi o σ(e0) is reducible, in which case ai ∈ β − 1
2 + Z≥0;
(3) νβτi o σ(e0) is reducible, in which case ai ∈ β + Z≥0;(4) νxτi o σ(e0) is irreducible for all x ∈ R
(a) τi ∼= τi but cd(τ) · σ(e0) 6∼= σ(e0) for SO2n, SO∗2n+2,
(b) τi ∼= τi but ωτσ(e0) 6∼= σ(e0) for GSp2n, GU2n+1, GU2n,
(c) τi ∼= τi but ωτ(cd(τ) · σ(e0)) 6∼= σ(e0) for GSO2n, GSO
∗2n+2,
(d) ν−2βωσ(e0) τi ∼= τi but cd(τ) · σ(e0) 6∼= σ(e0) for GSpin2n, GSpin∗2n+2,
in which case ai ∈ β + Z≥0. (This case does not occur for SO2n+1,Sp2n, U2n+1, U2n, GSpin2n+1.)
8 GENERIC REPRESENTATIONS FOR QUASI-SPLIT SIMILITUDE GROUPS
Then, if π is the generic subquotient of δ(∆1) × · · · × δ(∆k) o σ(e0), π isessentially square-integrable. Conversely, any generic irreducible essentiallysquare-integrable π of a group Gn(F ) is of this form (with ∆1, . . . ,∆k uniqueup to permutation), and further
π ↪→ δ(∆1)× · · · × δ(∆k) o σ(e0).
Note that the cusidal reducibility values in (1)–(3) follow from [Sha].
GENERIC REPRESENTATIONS FOR QUASI-SPLIT SIMILITUDE GROUPS 9
2.2. Generic essentially tempered representations.
Proposition. Let τ1, . . . , τc generic irreducible unitary supercuspidal repre-sentations of general linear groups and σ(e2) a generic irreducible essentiallysquare-integrable representation of Gn(F ). Let Ψ1, . . . ,Ψc be segments of the
form Ψi = [ν−ki+1
2 τi, νki−12 τi]. Then the generic component
σ(et) ↪→ νβδ(Ψ1)× · · · × νβδ(Ψc) o σ(e2)
is a generic essentially tempered representation, where β = β(σ(e2)). Further,any generic essentially tempered representation may be realized this way (withinducing representation unique up to Weyl conjugation).
This follows directly from a result of Harish-Chandra (cf. Proposition III.4.1[Wa]).
10 GENERIC REPRESENTATIONS FOR QUASI-SPLIT SIMILITUDE GROUPS
2.3. Generic admissible representations.
Notation.
• For Σ = [νaξ, νbξ], set Σ = [ν−bξ, ν−aξ] (so δ(Σ)∨ = δ(Σ)), whereˇ =contragredient composed with Galois conjugation for unitary andgeneral unitary groups and contragredient otherwise.
• ω′σ(et) =
{ωσ(et) (central character) for general spin groups,1 otherwise.
Note that by the standard module conjecture ([H-O]), the Langlands quo-tient of a generic standard module is generic if and only if the standardmodule induces irreducibly.
GENERIC REPRESENTATIONS FOR QUASI-SPLIT SIMILITUDE GROUPS 11
Theorem (the Langlands classification). 2 Let δ1, . . . , δk be essentially squareintegrable representations of general linear groups and and T an essentiallytempered representation of Gn(F ) satisfying
ε(δ1) ≥ · · · ≥ ε(δk) > β(T ).
Thenδ1 × · · · × δk o T
contains a unique irreducible quotient L(δ1⊗· · ·⊗ δk⊗T ). Further, any irre-ducible admissible representation may be written in this form, with the dataunique up to permutations among representations of general linear groupshaving the same central exponents.
See [B-W], [Sil], [Kon], etc., for the general result; [Tad1] (Sp2n and GSp2n),[Jan1] (SO2n+1 and SO2n), [KimW] (GSpin2n+1, GSpin2n, GSpin
∗2n).
2Note that this formulation requires an artifice for SO2n, GSpin2n, GSO2n (also used later).
12 GENERIC REPRESENTATIONS FOR QUASI-SPLIT SIMILITUDE GROUPS
Theorem. Suppose δ(Σi) = νxiδi, i = 1, 2, · · · , f with x1 ≥ x2 ≥ · · · ≥ xf >β. Then, the representation
δ(Σ1)× · · · × δ(Σj) o σ(et)
is irreducible if and only if {Σj}fj=1 and σ(et) satisfy the following properties:
(1) δ(Σi)× δ(Σj) and δ(Σi)×ω′σ(et)δ(Σj) are irreducible for all 1 ≤ i 6= j ≤f ; and
(2) δ(Σi) o σ(et) is irreducible for all 1 ≤ i ≤ f .
The reducibility for (1) is known from [Zel]; for (2) we write σ(et) as in theproposition above. Then δ(Σ)oσ(et) is irreducible if and only if the followinghold:
(1) δ(Σ)×νβδ(Ψj) and ω′σ(e2)δ(Σ)×νβδ(Ψj) are irreducible for all 1 ≤ j ≤ c;
and(2) δ(Σ) o σ(e2) is irreducible.
GENERIC REPRESENTATIONS FOR QUASI-SPLIT SIMILITUDE GROUPS 13
To understand when the second condition above holds, write σ(e2) as in thetheorem above. Then, δ(Σ) o σ(e2) is irreducible if and only if the followinghold:
(1) δ(Σ)×δ(∆i) and ω′σ(e0)δ(Σ)×δ(∆i) are irreducible for all i = 1, 2, · · · , k′;
and(2) either (a) δ(Σ) o σ(e0) is irreducible, or (b) δ(Σ) = δ([ν1+βξ, νb+βξ]),
with ν1+βξoσ(e0) reducible and there is some i having δ(∆i) = δ([ν1+βξ, νbi+βξ])and bi ≥ b.
Finally, for the second condition above, we have δ(Σ) o σ(e0) is irreducibleif and only if one of the following hold: for Σ = [ν−aξ, νbξ], we have
(1) ξ 6∼= ξ; or(2) ξ ∼= ξ and the following: (i) if νxξ o σ(e0) is reducible for some (nec-
essarily unique) x = α ≥ 0, then ±α 6∈ {−a,−a + 1, · · · , b}; (ii) ifνxξ o σ(e0) is irreducible for all x ≥ 0, then a /∈ Z≥0.
14 GENERIC REPRESENTATIONS FOR QUASI-SPLIT SIMILITUDE GROUPS
3. Tools
The main results needed:
• results on general linear groups• cuspidal reducibility conditions• standard module conjecture• the Langlands classification and Casselman criterion• formula for twisting an induced representation by characters• µ∗ structure formula for calculating Jacquet modules
GENERIC REPRESENTATIONS FOR QUASI-SPLIT SIMILITUDE GROUPS 15
3.1. Twisting by characters. The formulas below are analogues of thatfor GSp2n in [S-T].
Lemma. Let χ be a character of F× and identify χ with a character of Gn(F )via composition with the similitude character.
(1) For Gn = GSpin2n+1, χ(π o θ) ∼={χπ o χθ if n > 0χπ o χ2θ if n = 0.
(2) For Gn = GSp2n, GSO∗2n+2, GU2n+1, GU2n, χ(π o θ) ∼= π o χθ.(3) For Gn = GSO2n, χ(π o θ) ∼= π o χθ (n 6= 1).
(4) For Gn = GSpin2n, χ(π o θ) ∼={χπ o χθ if n > 1,χπ o χ2θ if n = 0.
(5) For Gn = GSpin∗2n+2, χ(π o θ) ∼= χπ o χθ.
16 GENERIC REPRESENTATIONS FOR QUASI-SPLIT SIMILITUDE GROUPS
3.2. µ∗ structure formula. This has been done in [Tad2] (SO2n+1, Sp2n,and GSp2n), [M-T] (U2n+1 and U2n), [J-L] (SO2n), [KimY1] (GSpin2n+1),[KimY2] (GSpin2n), [K-M] (GU2n). The corresponding result for general lin-ear groups is in [Zel]. Note that this also uses the artifice for SO2n, GSpin2n,GSO2n.
Definition. For general linear groups,
m∗ =n∑k=0
rMk,G,
where Mk = GLk ×GLn−k. For Gn, we set
µ∗ =n∑k=0
rMk,G,
where Mk = GLk ×Gn−k.
GENERIC REPRESENTATIONS FOR QUASI-SPLIT SIMILITUDE GROUPS 17
LetN ∗ = ( ⊗m∗)D ◦ s ◦m∗,
where s : τ1 ⊗ τ2 7→ τ2 ⊗ τ1 and
( ⊗m∗)D(τ1 ⊗ τ2) =
{τ1 ⊗m∗(τ2)⊗ e if τ1 a representation of GLeven,τ1 ⊗m∗(τ2)⊗ c if τ1 a representation of GLodd.
18 GENERIC REPRESENTATIONS FOR QUASI-SPLIT SIMILITUDE GROUPS
Theorem. 3
µ∗(τ o π) = N ∗(τ)oµ∗(π),
where
(ρ1 ⊗ ρ2 ⊗ ρ3 ⊗ d)o(ρ⊗ σ)
=
(ρ1 × ρ2 × ρ)⊗ (ρ3 o σ) for Gn = SO2n+1, Sp2n, U2n+1, U2n,(ωσρ1 × ρ2 × ρ)⊗ (ρ3 o σ) for Gn = GSpin2n+1,(ρ1 × ρ2 × ρ)⊗ (ρ3 o ωρ1σ) for Gn = GSp2n, GU2n+1, GU2n,(ρ1 × ρ2 × ρ)⊗ d(ρ3 o σ) for Gn = SO2n, SO
∗2n+2,
(ρ1 × ρ2 × ρ)⊗ d(ρ3 o ωρ1σ) for Gn = GSO2n, GSO∗2n+2,
(ωσρ1 × ρ2 × ρ)⊗ d(ρ3 o σ) for Gn = GSpin2n, GSpin∗2n+2.
3Thanks to [Arc] for helping us realize the role of c in the quasi-split case.
GENERIC REPRESENTATIONS FOR QUASI-SPLIT SIMILITUDE GROUPS 19
References
[Arc] S. Archava, personal communication.[Asg] M. Asgari, Local L-functions for split spinor groups, Canad. J. Math., 54(2002), no. 4, 673-693.[ACS] M. Asgari, J. Cogdell, and F. Shahidi, Local transfer and reducibility of induced representations
of p-adic groups of classical type, in Advances in the Theory of Automorphic Forms and TheirL-functions, 1-22, Contemp. Math., 664(2016), American Mathematical Society, Providence, RI.
[B-W] A. Borel and N. Wallach, Continuous Cohomology, Discrete Subgroups, and Representations ofReductive Groups, Princeton University Press, Princeton, 1980.
[Cas] W. Casselman, The structure of reductive groups, preprint.[Gol] D. Goldberg, R-groups and elliptic representations for similitude groups, Math. Ann., 307(1997),
no. 4, 569-588.[H-O] V. Heiermann and E. Opdam, On the tempered l-functions conjecture, Amer. J. Math., 135(2013),
no. 3, 777-799.[H-S] J. Hundley and E. Sayag, Descent construction for GSpin groups, Mem. Amer. Math. Soc.,
1148(2016), 1-125.[Jac] H. Jacquet, Generic representations. Non-commutative harmonic analysis (Actes Colloq., Marseille-
Luminy, 1976), pp. 91–101. Lecture Notes in Math., Vol. 587, Springer, Berlin, 1977.[Jan1] C. Jantzen, Degenerate principal series for orthogonal groups, J. reine angew. Math., 441(1993),
61-98.[Jan2] C. Jantzen, On supports of induced representations for symplectic and odd-orthogonal groups,
Amer. J. Math., 119(1997), no. 6, 1213–1262.[J-L] C. Jantzen and B. Liu, The generic dual of p-adic split SO2n and local Langlands parameters, Israel
J. Math., 204(2014), 199-260.
20 GENERIC REPRESENTATIONS FOR QUASI-SPLIT SIMILITUDE GROUPS
[J-S] D. Jiang and D. Soudry, Generic representations and local Langlands reciprocity law for p-adicSO2n+1, in Contributions to Automorphic Forms, Geometry, and Number Theory, 457–519, JohnsHopkins University Press, Baltimore, MD, 2004.
[KimW] W. Kim, Square integrable representations and the standard module conjecture for general spingroups, Canad. J. Math., 61(2009), no. 3, 617-640.
[KimY1] Y. Kim, Strongly positive representations of GSpin2n+1 and the Jacquet module method, Math.Z., 279(2015), no. 1-2, 271-296.
[KimY2] Y. Kim, Strongly positive representations of even spin groups, Pacific J. Math., 280(2016), no. 1,69-88.
[K-M] Y. Kim and I. Matic, Classification of strongly positive representations of even general unitarygroups, preprint.
[Kon] T. Konno, A note on the Langlands classification and irreducibility of induced representations ofp-adic groups. Kyushu J. Math., 57(2003), 383-409
[Liu] B. Liu, Genericity of representations of p-adic Sp2n and local Langlands parameters, Canad. J.Math., 63(2011), no. 5, 1107–1136.
[M-T] C. Mœglin and M. Tadic, Construction of discrete series for classical p-adic groups, J. Amer. Math.Soc. 15(2002), no. 3, 715-786.
[Mui] G. Muic, On generic irreducible representations of Sp(n,F) and SO(2n+1,F), Glas. Mat. Ser. III,33(53) (1998), no. 1, 19–31.
[S-T] P. Sally and M. Tadic, Induced representations and classification for GSp(2, F ) and Sp(2, F ), Mem.Soc. Math. France, 52(1993), 75-133.
[Sha] F. Shahidi, A proof of Langlands conjecture on Plancherel measure; complementary series for p-adicgroups Ann. of Math., 132(1990), no. 2, 273-330.
[Sil] A. Silberger, The Langlands quotient theorem for p-adic groups., Math. Ann., 236(1978), 95-104.[Tad1] M. Tadic, Representations of p-adic symplectic groups, Compositio Math., 90(1994), no. 2, 123-181.
GENERIC REPRESENTATIONS FOR QUASI-SPLIT SIMILITUDE GROUPS 21
[Tad2] M. Tadic, Structure arising from induction and Jacquet modules of representations of classicalp-adic groups, J. Algebra, 177(1995), no. 1, 1-33.
[Tad3] M. Tadic, On reducibility of parabolic induction, Israel J. Math., 107(1998), 29–91.[Wa] J.-L. Waldspurger, La formule de Plancherel pour les groupes p-adiques d’apres Harish-Chandra,
J. Inst. Math. Jussieu, 2(2003), no. 2, 235-333.[Xu] B. Xu, L-packets of quasisplit GSp(2n) and GO(2n), preprint.[Zel] A. V. Zelevinsky, Induced representations of reductive p-adic groups. II. On irreducible representa-
tions of GL(n). Ann. Sci. Ecole Norm. Sup. (4)13 (1980), no. 2, 165–210.
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