Fourier Coefficients and Automorphic Discrete Spectrum of Classical Groups. Dihua Jiang University of Minnesota
|
|
- Morris James
- 6 years ago
- Views:
Transcription
1 Fourier Coefficients and Automorphic Discrete Spectrum of Classical Groups Dihua Jiang University of Minnesota KIAS, Seoul November 16, 2015
2 Square-Integrable Automorphic Forms G a reductive algebraic group defined over a number field F. A is the ring of adeles of F. X G := G(F )\G(A) 1, where G(A) 1 := χ X (G) ker χ A. L 2 (X G ) denotes the space of functions: φ : X G C such that G(Q)\G(A) 1 φ(g) 2 dg <. A 2 (G) is the set of equivalence classes of irreducible unitary representations of G(A) occurring in the discrete spectrum L 2 disc (X G). A cusp (G) is the subset of A 2 (G) consisting of those automorphic representations of G(A) occurring in the cuspidal spectrum L 2 cusp(x G ).
3 Theory of Endoscopic Classification Theorem Let G be an F -quasisplit classical group and G be a pure inner form of G over F. For any π A cusp (G), there is a global Arthur parameter ψ Ψ 2 (G ), which is G-relevant, such that π Π ψ (G) where Π ψ (G) is the global Arthur packet of G associated to ψ. G is a quasisplit SO or a split Sp, Arthur s book 2013, including Chapter 9 for non-quasisplit case. G is a quasiplit U m, C.-P. Mok Mem. of AMS, G is GSp and GSO, Bin Xu (2014, 2015) (student of Arthur). G is a pure inner form of quasisplit U m, preprint of 2014 by Kaletha-Minguez-Shin-White. For more general unitary groups, it is their work in progress.
4 Global Arthur Parameters Ψ 2 (G): Examples G = SO 2n+1, F -split, and (G ) = Sp 2n (C). Each ψ Ψ 2 (G ) (global Arthur parameters) is written as a formal sum of simple Arthur parameters: ψ = ψ 1 ψ 2 ψ r where ψ i = (τ i, b i ), with τ i A cusp (GL ai ); a i, b i 1; and r i=1 a ib i = 2n. If i j, either τ i = τj or b i b j, with the parity condition that a i b i is even and ψ i Ψ 2 (SO a i b i +1 ). Endoscopy Structure: 2n = r i=1 a i b i, SO a 1 b 1 +1 SO a r b r+1 = SO 2n+1 Π ψ1 ( ) Πψr ( ) = Π ψ ( )
5 Global Arthur Parameters Ψ 2 (G): Examples G = Sp 2n, F -split, and (G ) = SO 2n+1 (C). Each ψ Ψ 2 (G ) is written as a formal sum of simple Arthur parameters: ψ = ψ 1 ψ 2 ψ r where ψ i = (τ i, b i ), with τ i A cusp (GL ai ); a i, b i 1; r i=1 a ib i = 2n + 1; and r i=1 ωb i τ i = 1. If i j, either τ i = τj or b i b j, with the parity: 1 If a i b i is even, then ψ i Ψ 2 (SO a ib i ); 2 If a i b i is is odd, then ψ i Ψ 2 (Sp a ib i 1). Endoscopy Structure: 2n + 1 = r i=1 a i b i, a i b i =2l i SO 2l i a j b j =2l j +1 Sp 2l j = Sp 2n ai b i =2l i Πψi ( ) aj b j =2l j +1 Π ψj ( ) = Π ψ ( )
6 Global Arthur Parameters Ψ 2 (G): Examples A parameter ψ = ψ 1 ψ 2 ψ r Ψ 2 (G ) is generic if b 1 = = b r = 1. Generic global Arthur parameters φ Φ 2 (G ) are: φ = (τ 1, 1) (τ 2, 1) (τ r, 1) with τ i A cusp (GL ai ) that τ i = τj if i j. They are of either symplectic or orthogonal type, depending on G. The pure inner forms of G = SO m are G = SO m (V, q) for non-deg. quad. spaces (V, q) over F with the same dimension and discriminant. If G is a pure inner form of G, then L G = L G. For φ Φ 2 (G ), the endoscopic classification may define the global Arthur packet Π φ (G ) and also define the global Arthur packet Π φ (G), which is non-empty if φ is G-relevant.
7 Endoscopic Classification and Langlands Functoriality Ψ 2 (G ) G ψ (1) A 2 (G) Π ψ (G) Πψ (G ) A 2 (G ) Remarks: In W. Schmid s 70th birthday conference, Arthur explains that the stable trace formula method obtains the character relations among those objects. Arthur-Schmid Question: How to construct modules for π in A 2 (G) Π ψ (G) in general?
8 Problems Based on the Endoscopic Classification By using the automorphic descent of Ginzburg-Rallis-Soudry, one can show that Π φ (G ) A cusp (G ). Problem (1): What happens if G is replaced by general G? I addressed this problem in my lecture at the Roger Howe 70th birthday conference, which covers mainly my joint work with Lei Zhang, and answers a Question of Arthur-Schmid on explicit constructions of automorphic modules for cuspidal automorphic representations in terms of their global Arthur parameters. The goal is to construct all members in Π φ (G) A cusp (G). Problem (2): What happens if the generic Arthur parameters φ are replaced by general Arthur parameters ψ? This is part of my joint project with Baiying Liu, and will be the main focus of my lecture today.
9 Problems Based on the Endoscopic Classification First of all, for classical groups G and generic Arthur parameters φ in Φ 2 (G ), which is G-relevant, one can show that Π φ (G) A 2 (G) is contained in A cusp (G). However, for a general Arthur parameter ψ, the set Π ψ (G) A cusp (G) may be empty, even if the set Π ψ (G) A 2 (G) is not empty. Hence we are interested in the conditions on ψ that the set Π ψ (G) A cusp (G) is empty; and if it is not empty, we want to construct members in the set Π ψ (G) A cusp (G). The idea is to use Fourier coefficients of automorphic representations to determine their automorphic wave-front set. This leads to its connections with the notion of singular automorphic forms in terms of the representation-theoretic formulation of Roger Howe and the work of Jianshu Li.
10 Fourier Coefficients and Nilpotent Adjoint Orbits G is an F -quasi-split classical group and g is the Lie algebra. Let N G be dimension for the defining embedding G GL(N G ). Over algebraic closure F of F, all the nilpotent elements in g (F ) form a conic algebraic variety, called the nilcone N (g ). Under the adjoint action of G, N (g ) decomposes into finitely many adjoint G -orbits O, which are parameterized by the corresponding partitions of N = N G of type G. Over F, each F -orbit reduces to an F -stable adjoint G (F )-orbits O st, and hence the F -stable adjoint orbits in the nilcone N (g ) are also parameterized by the corresponding partitions of an integer N = N G of type G.
11 Fourier Coefficients and Nilpotent Adjoint Orbits For X N (g ), use sl 2 -triple (over F ) to define a unipotent subgroup V X and a character ψ X. Let {X, H, Y } be an sl 2 -triple (over F ). Under the adjoint action of ad(h), g = g r g 2 g 1 g 0 g 1 g 2 g r. Ad(G )(Y ) g 2 and Ad(G )(X) g 2 are Zariski dense in g 2 and g 2, respectively. Take V X to be the unipotent subgroup of G such that the Lie algebra of V X is equal to i 2 g i. Let ψ F be a non-trivial additive character of F \A. The character ψ X of V X (F ) or V X (A) is defined by ψ X (v) = ψ F (tr(y log(v))).
12 Fourier Coefficients and Nilpotent Adjoint Orbits The Fourier coefficient of ϕ π A 2 (G ) is defined by F ψ X (ϕ)(g) := ϕ(vg)ψ X (v) 1 dv. V X (F )\V X (A) Since ϕ is automorphic, the nonvanishing of F ψ X (ϕ) depends only on the G (F )-adjoint orbit O X of X. The set n(ϕ) := {X N (g) F ψ X (ϕ) 0} is stable under the G (F )-adjoint action. Denoted by p(ϕ) the set of partitions p of N G of type G corresponding to the F -stable orbits Op st that have non-empty intersection with n(ϕ). p m (ϕ) is the set of all maximal partitions in p(ϕ), according to the partial ordering of partitions.
13 Maximal Fourier Coefficients of Automorphic Forms For π A 2 (G), denote by p m (π) the set of maximal members among p m (ϕ) for all ϕ π. One may think of the set p m (π) as an algebraic version of the wave-front set of π in the sense of Roger Howe. Basic Problems: How to determine the set p m (π) in terms of other invariants of π? What can one say about π if one knows the structure of p m (π)? Folklore Conjecture: For any irreducible automorphic representation π of G, the set p m (π) contains only one partition.
14 Maximal Fourier Coefficients of Automorphic Forms Examples: G = GL n, the G(F )-stable orbits in N (g) are parameterized by partitions of n. Theorem (Piatetski-Shapiro; Shalika): If π A 2 (GL n ) is cuspidal, p m (π) = {[n]}. This says that any irreducible cuspidal automorphic representation has a nonzero Whittaker-Fourier coefficient. What happens if π A 2 (GL n ) is not cuspidal? Moeglin-Waldspurger Theorem: Any π A disc (GL n ) has form (τ, b) (Speh residue with cuspidal support τ b ), where τ A cusp (GL a ) and n = ab. If π = (τ, b), then p m (π) = {[a b ]} (Ginzburg(2006), J.-Liu(2013) gives a complete global proof). In particular, the Folklore Conjecture is verified for all π A 2 (GL n )!
15 Maximal Fourier Coefficients and Arthur Parameters How to understand this in terms of Arthur parametrization? (τ, b) has the Arthur parameter ψ = (τ, b). The partition attached to ψ is p ψ := [b a ] and p m ( (τ, b)) = {[a b ]}. η([b a ]) = [a b ] is given by the Barbasch-Vogan duality η from GL n to GL n. Take an Arthur parameter for GL n : for τ i A cusp (GL ai ), ψ = (τ 1, b 1 ) (τ 2, b 2 ) (τ r, b r ). The partition attached to ψ is p ψ = [b a 1 1 ba 2 2 bar r ]. The Arthur representation is an isobaric sum π ψ = (τ 1, b 1 ) (τ 2, b 2 ) (τ r, b r ). Conjecture: p m (π ψ ) = {η gl n,gl n (p ψ )}.
16 Maximal Fourier Coefficients and Arthur Parameters For ψ = ψ 1 ψ 2 ψ r Ψ 2 (G ), where ψ i = (τ i, b i ) with τ i A cusp (GL ai ) and b i 1, p ψ = [b a 1 1 bar r ] is the partition of N G attached to (ψ, (G ) ) and η(p ψ ) is the Barbasch-Vogan duality of p ψ from (G ) to G. Conjecture (J.-2014): (1) For every π Π ψ (G ) A 2 (G ), any partition p p m (π) has the property that p η(p ψ ). (2) There exists at least one member π Π ψ (G ) A 2 (G ) having the property that η(p ψ ) p m (π). Remark: For a pure inner form G of G, assume that the global Arthur parameter ψ is G-relevant and the Barbasch-Vogan duality η(p ψ ) is a G-relevant partition of N G = N G of type G. Then the above Conjecture should also hold for G and G-relevant Arthur parameters.
17 Examples of the Barbasch-Vogan duality G = SO 2n+1 and 2n = ab; Take ψ = (τ, b) for τ A cusp (GL a ), and { 2l, if τ is orthogonal, b = 2l + 1, if τ is symplectic. p ψ = [b a ] is the partition of 2n of type (ψ, Sp 2n (C)). The Barbasch-Vogan duality is given as follows: [(a + 1)a b 2 (a 1)1] if b = 2l and a is even; η(p ψ ) = [a b 1] if b = 2l and a is odd; [(a + 1)a b 1 ] if b = 2l + 1.
18 Examples of the Barbasch-Vogan duality Take G = Sp 2n and ψ = (τ, 2b + 1) r i=2 (τ i, 1) Ψ 2 (G). p ψ = [(2b + 1) a (1) 2m+1 a ] with 2m + 1 = (2n + 1) 2ab. When a 2m and a is even, η(p ψ ) =η([(2b + 1) a (1) 2m+1 a ]) = [(2b + 1) a (1) 2m a ] t =[(a) 2b+1 ] + [(2m a)] = [(2m)(a) 2b ]. When a 2m and a is odd, η(p ψ ) =η([(2b + 1) a (1) 2m+1 a ]) =([(2b + 1) a (1) 2m a ] Sp2n ) t =[(2b + 1) a 1 (2b)(2)(1) 2m a 1 ] t =[(a 1) 2b+1 ] + [(1) 2b ] + [(1) 2 ] + [(2m 1 a)] =[(2m)(a + 1)(a) 2b 2 (a 1)].
19 Remarks on the Conjecture It is true when G = GL n and ψ is an Arthur parameter for the discrete spectrum. If φ Φ 2 (G) is generic, i.e. b 1 = = b r = 1, the partition p φ = [1 N G]. The Barbasch-Vogan duality of p φ is η([1 N G]) = [N G ] G. It is clear that the partition η([1 N G]) is G-relevant if G = G is quasi-split. The conjecture claims that any generic global Arthur packet contains a generic member for quasi-split G, and hence implies the global Shahidi conjecture on genericity of tempered packets. This special case can be proved by the Arthur-Langlands transfer from G to GL NG and the Ginzburg-Rallis-Soudry descent.
20 Remarks on the Conjecture The conjecture has been verified for various cases of Sp 2n through my joint work with Liu. The conjecture provides an upper bound partition for π Π ψ (G) A 2 (G), with a given Arthur parameter ψ. It is our work in progress to obtain a lower bound partition for π Π ψ (G) A 2 (G), with a given Arthur parameter ψ. It is very interesting, but harder problem to determine p m (π) for general automorphic members π Π ψ (G) A 2 (G) for each given Arthur parameter ψ! The theory of singular automorphic forms of Howe and Li provides a lower bound partition for π A cusp (G). It is not hard to find that the lower bound partition provided by the theory of Howe and Li may not be the best for all π A cusp (G).
21 Singular Partitions and Rank in the Sense of Howe G = Sp 2n. Take P n = M n U n to be the Siegel parabolic of Sp 2n, with M n = GLn and the elements of U n are of form ( ) In X u(x) =. 0 I n By the Pontryagin duality, one has U n (F )\U n (A) = Sym 2 (F n ). For a fixed nontrivial additive character ψ F of F \A, a T Sym 2 (F n ) corresponds to the character ψ T by ψ T (u(x)) := ψ F (tr(t wx)), with w anti-diagonal, and the action of GL n on Sym 2 (F n ) is induced from its adjoint action on U n.
22 Singular Partitions and Rank in the Sense of Howe For an automorphic form ϕ on Sp 2n (A), the T - or ψ T -Fourier coefficient is defined by F ψ T (ϕ)(g) := ϕ(u(x)g)ψ 1 T (u(x))du(x). U m(f )\U n(a) An automorphic form ϕ on Sp 2n (A) is called non-singular if ϕ has a nonzero ψ T -Fourier coefficient with a non-singular T. ϕ is called singular if ϕ has the property that if a nonzero ψ T -Fourier coefficient F ψ T (ϕ) is nonzero, then det(t ) = 0. Howe shows in 1981 that if an automorphic form ϕ on Sp 2n (A) is singular, then ϕ can be expressed as a linear combination of certain theta functions. Jianshu Li shows in 1989 that any cuspidal automorphic form of Sp 2n (A) is non-singular.
23 Singular Partitions and Rank in the Sense of Howe For a split SO m defined by a non-deg. quad. space (V, q) over F of dim m with the Witt index [ m 2 ], let X+ be a maximal totally isotropic subspace of V with dim [ m 2 ] and let X be the dual to X + : V = X + V 0 + X + with V 0 the orth. complement of X + X + (dim V 0 1). The generalized flag {0} X + V defines a maximal parabolic subgroup P X +, whose Levi part M X + is isomorphic to GL [ m 2 ] and whose unipotent radical N X + is abelian if m is even; and is a two-step unipotent subgroup with its center Z X + if m is odd. When m is even, we set Z X + = N X +.
24 Singular Partitions and Rank in the Sense of Howe By the Pontryagin duality, For any T 2 (F [ m 2 ] ), Z X +(F )\Z X +(A) = 2 (F [ m 2 ] ). ψ T (z(x)) := ψ F (tr(t wx)). For an automorphic form ϕ on G(A), the T or ψ T Fourier coefficient is defined by F ψ T (ϕ)(g) := ϕ(z(x)g)ψ 1 T (z(x))dz(x). Z X + (F )\Z X + (A) An automorphic form ϕ on G(A) is called non-singular if ϕ has a non-zero ψ T -Fourier coefficient with T 2 (F [ m 2 ] ) of maximal rank.
25 Singular Partitions and Rank in the Sense of Howe Denote by p ns the partition corresponding to the non-singular Fourier coefficients. For Sp 2n, p ns = [2 n ], which is a special partition for Sp 2n. For SO 2n+1, one has p ns = { [2 2e 1] if n = 2e; [2 2e 1 3 ] if n = 2e + 1. This is not a special partition of SO 2n+1. For SO 2n, one has p ns = { [2 2e ] if n = 2e; [2 2e 1 2 ] if n = 2e + 1. This is a special partition of SO 2n.
26 Singular Partitions and Lower Bound Partitions J.-Liu-Savin show 2015 that for any automorphic representation π, the set p m (π) contains only special partitions. For SO 2n+1, p ns p m (π) for any π A cusp (SO 2n+1 ). Any p p m (π) as π runs in A cusp (SO 2n+1 ) must be bigger than or equal to the following partition { [32 2e ] if n = 2e; p Gn ns p SO 2n+1 ns = [32 2e ] if n = 2e + 1. denotes the G n -expansion of the partition p ns, i.e., the smallest special partition which is bigger than or equal to p ns. Proposition: For a split classical group G n, p Gn is a lower ns bound for partitions in the set p m (π) as π runs in A cusp (G n ).
27 Singular Partitions and Small Cuspidal Representations It is natural to ask whether the lower bound p Gn ns is sharp? For n = 2e even, and F to be totally real, the Ikeda lifting gives π A cusp (Sp 4e ) with the global Arthur parameter (τ, 2e) (1, 1), where τ A cusp (GL 2 ) is self-dual and has the trivial central character. First, for any p p m (π), one has p [2 2e ] = p Sp 4e. ns Then, one also has p Sp 4e = [2 2e ] p. ns Proposition: If F is totally real, the Ikeda lifting π of Sp 4e has: p Sp 4e = [2 2e ] p, and hence the non-singular partition ns p Sp 4e = p ns = [2 2e ] is the sharp lower bound for A cusp (Sp 4e ). ns Note that the construction of Ikeda lifting does not work when F is not totally real or n = 2e + 1 is odd. What happens in such situations?
28 Singular Partitions and Small Cuspidal Representations J.-Liu (2015): If F is totally imaginary and n 5, the global Arthur packet Π (τ,n) (ɛ,1) (Sp 2n ), with τ A cusp (GL 2 ) self-dual, has no cuspidal members, where ɛ = 1 if n = 2e; and ɛ = ω τ if n = 2e + 1. In this case, the construction of the Ikeda lifting is impossible! Theorem (J.-Liu 2015): 1 For any π A cusp (Sp 2n ), p m (π) = [2 n ] if and only if π is hypercuspidal in the sense of I. Piatetski-Shapiro. 2 For ψ = (τ, 2e) (1, 1) Ψ 2 (Sp 4e ) with τ A cusp (GL 2 ), any cuspidal π Π ψ (Sp 4e ) has that p m (π) = [2 2e ]. 3 For ψ = (τ, 2e + 1) (ω τ, 1) Ψ 2 (Sp 4e+2 ) with τ A cusp (GL 2 ), any cuspidal π Π ψ (Sp 4e+2 ) has that p m (π) = [2 2e+1 ]. 4 For ψ = (τ, 2e + 1) Ψ 2 (Sp 6e+2 ) with τ A cusp (GL 3 ), any cuspidal π Π ψ (Sp 6e + 2) has that p m (π) = [2 3e+1 ].
29 Singular Partitions and Small Cuspidal Representations Theorem (J.-Liu 2015): Assume that F is totally imaginary, and Part (1)of the Conjecture holds. For ψ = r i=1(τ i, b i ) Ψ 2 (Sp 2n ) with τ i A cusp (GL ai ) for i = 1, 2,, r, and 2n = ( r a i b i ) 1, i=1 then there exists a constant N 0 depending on (a 1,, a r ) and (b 1,, b r ), such that if 2n > N 0, the global Arthur packet Π ψ (Sp 2n ) contains no cuspidal member.
30 Singular Partitions and Small Cuspidal Representations Example: Take ψ = (1, b 1 ) (τ, b 2 ) with b 1 1 odd, and with τ A cusp (GL 2 ) of symplectic type and b 2 even. Then N 0 = 8 and the global Arthur packet Π ψ (Sp ) contains no cuspidal members except that (b 1, b 2 ) = (1, 2), (1, 4), (3, 2), or (5, 2). Work in Progress: Characterize and construct small cuspidal representations in general. THANKS!
On Cuspidal Spectrum of Classical Groups
On Cuspidal Spectrum of Classical Groups Dihua Jiang University of Minnesota Simons Symposia on Geometric Aspects of the Trace Formula April 10-16, 2016 Square-Integrable Automorphic Forms G a reductive
More informationOn the Non-vanishing of the Central Value of Certain L-functions: Unitary Groups
On the Non-vanishing of the Central Value of Certain L-functions: Unitary Groups arxiv:806.04340v [math.nt] Jun 08 Dihua Jiang Abstract Lei Zhang Let π be an irreducible cuspidal automorphic representation
More informationOn the Genericity of Cuspidal Automorphic Forms of SO 2n+1
On the Genericity of Cuspidal Automorphic Forms of SO 2n+1 Dihua Jiang School of Mathematics University of Minnesota Minneapolis, MN55455, USA David Soudry School of Mathematical Sciences Tel Aviv University
More informationEndoscopic character relations for the metaplectic group
Endoscopic character relations for the metaplectic group Wen-Wei Li wwli@math.ac.cn Morningside Center of Mathematics January 17, 2012 EANTC The local case: recollections F : local field, char(f ) 2. ψ
More informationThe Shimura-Waldspurger Correspondence for Mp(2n)
The Shimura-Waldspurger Correspondence for Wee Teck Gan (Joint with Atsushi Ichino) April 11, 2016 The Problem Let 1 µ 2 Sp(2n) 1 be the metaplectic group (both locally and globally). Consider the genuine
More informationON THE RESIDUAL SPECTRUM OF SPLIT CLASSICAL GROUPS SUPPORTED IN THE SIEGEL MAXIMAL PARABOLIC SUBGROUP
ON THE RESIDUAL SPECTRUM OF SPLIT CLASSICAL GROUPS SUPPORTED IN THE SIEGEL MAXIMAL PARABOLIC SUBGROUP NEVEN GRBAC Abstract. For the split symplectic and special orthogonal groups over a number field, we
More informationWeyl Group Representations and Unitarity of Spherical Representations.
Weyl Group Representations and Unitarity of Spherical Representations. Alessandra Pantano, University of California, Irvine Windsor, October 23, 2008 β ν 1 = ν 2 S α S β ν S β ν S α ν S α S β S α S β ν
More informationPrimitive Ideals and Unitarity
Primitive Ideals and Unitarity Dan Barbasch June 2006 1 The Unitary Dual NOTATION. G is the rational points over F = R or a p-adic field, of a linear connected reductive group. A representation (π, H)
More informationA PRODUCT OF TENSOR PRODUCT L-FUNCTIONS OF QUASI-SPLIT CLASSICAL GROUPS OF HERMITIAN TYPE
A PRODUCT OF TENSOR PRODUCT L-FUNCTIONS OF QUASI-SPLIT CLASSICAL GROUPS OF HERMITIAN TYPE DIHUA JIANG AND LEI ZHANG Abstract. A family of global zeta integrals representing a product of tensor product
More informationRemarks on the Gan-Ichino multiplicity formula
SIMONS SYMPOSIA 2016 Geometric Aspects of the Trace Formula Sch oss ma A i - Remarks on the Gan-Ichino multiplicity formula Wen-Wei Li Chinese Academy of Sciences The cover picture is taken from the website
More informationRepresentation Theory & Number Theory Workshop April April 2015 (Monday) Room: S
Programme 20 April 2015 (Monday) Room: S17-04-05 9.30am 10.30am 10.30am 11.00am Local and global wave-front sets Gordan SAVIN, University of Utah 11.00am 12.00pm Local symmetric square L-functions and
More informationSPHERICAL UNITARY REPRESENTATIONS FOR REDUCTIVE GROUPS
SPHERICAL UNITARY REPRESENTATIONS FOR REDUCTIVE GROUPS DAN CIUBOTARU 1. Classical motivation: spherical functions 1.1. Spherical harmonics. Let S n 1 R n be the (n 1)-dimensional sphere, C (S n 1 ) the
More informationResults from MathSciNet: Mathematical Reviews on the Web c Copyright American Mathematical Society 1998
98m:11125 11R39 11F27 11F67 11F70 22E50 22E55 Gelbart, Stephen (IL-WEIZ); Rogawski, Jonathan (1-UCLA); Soudry, David (IL-TLAV) Endoscopy, theta-liftings, and period integrals for the unitary group in three
More informationIMAGE OF FUNCTORIALITY FOR GENERAL SPIN GROUPS
IMAGE OF FUNCTORIALITY FOR GENERAL SPIN GROUPS MAHDI ASGARI AND FREYDOON SHAHIDI Abstract. We give a complete description of the image of the endoscopic functorial transfer of generic automorphic representations
More informationSOME REMARKS ON REPRESENTATIONS OF QUATERNION DIVISION ALGEBRAS
SOME REMARKS ON REPRESENTATIONS OF QUATERNION DIVISION ALGEBRAS DIPENDRA PRASAD Abstract. For the quaternion division algebra D over a non-archimedean local field k, and π an irreducible finite dimensional
More informationReducibility of generic unipotent standard modules
Journal of Lie Theory Volume?? (??)???? c?? Heldermann Verlag 1 Version of March 10, 011 Reducibility of generic unipotent standard modules Dan Barbasch and Dan Ciubotaru Abstract. Using Lusztig s geometric
More informationGeometric Structure and the Local Langlands Conjecture
Geometric Structure and the Local Langlands Conjecture Paul Baum Penn State Representations of Reductive Groups University of Utah, Salt Lake City July 9, 2013 Paul Baum (Penn State) Geometric Structure
More informationOn Certain L-functions Titles and Abstracts. Jim Arthur (Toronto) Title: The embedded eigenvalue problem for classical groups
On Certain L-functions Titles and Abstracts Jim Arthur (Toronto) Title: The embedded eigenvalue problem for classical groups Abstract: By eigenvalue, I mean the family of unramified Hecke eigenvalues of
More informationLECTURE 2: LANGLANDS CORRESPONDENCE FOR G. 1. Introduction. If we view the flow of information in the Langlands Correspondence as
LECTURE 2: LANGLANDS CORRESPONDENCE FOR G J.W. COGDELL. Introduction If we view the flow of information in the Langlands Correspondence as Galois Representations automorphic/admissible representations
More informationTHE GROSS-PRASAD CONJECTURE AND LOCAL THETA CORRESPONDENCE. 1. Introduction
THE GROSS-RASAD CONJECTURE AND LOCAL THETA CORRESONDENCE WEE TECK GAN AND ATSUSHI ICHINO 1. Introduction In [15], [16], [9], [1], a restriction problem in the representation theory of classical groups
More informationCUBIC UNIPOTENT ARTHUR PARAMETERS AND MULTIPLICITIES OF SQUARE INTEGRABLE AUTOMORPHIC FORMS
CUBIC UNIPOTNT ARTHUR PARAMTRS AND MULTIPLICITIS OF SQUAR INTGRABL AUTOMORPHIC FORMS W TCK GAN, NADYA GURVICH AND DIHUA JIANG 1. Introduction Let G be a connected simple linear algebraic group defined
More informationReport on the Trace Formula
Contemporary Mathematics Report on the Trace Formula James Arthur This paper is dedicated to Steve Gelbart on the occasion of his sixtieth birthday. Abstract. We report briefly on the present state of
More informationTRILINEAR FORMS AND TRIPLE PRODUCT EPSILON FACTORS WEE TECK GAN
TRILINAR FORMS AND TRIPL PRODUCT PSILON FACTORS W TCK GAN Abstract. We give a short and simple proof of a theorem of Dipendra Prasad on the existence and non-existence of invariant trilinear forms on a
More informationA NOTE ON REAL ENDOSCOPIC TRANSFER AND PSEUDO-COEFFICIENTS
A NOTE ON REAL ENDOSCOPIC TRANSFER AND PSEUDO-COEFFICIENTS D. SHELSTAD 1. I In memory of Roo We gather results about transfer using canonical factors in order to establish some formulas for evaluating
More informationON THE RESIDUAL SPECTRUM OF HERMITIAN QUATERNIONIC INNER FORM OF SO 8. Neven Grbac University of Rijeka, Croatia
GLASNIK MATEMATIČKI Vol. 4464)2009), 11 81 ON THE RESIDUAL SPECTRUM OF HERMITIAN QUATERNIONIC INNER FORM OF SO 8 Neven Grbac University of Rijeka, Croatia Abstract. In this paper we decompose the residual
More informationCOHOMOLOGY OF ARITHMETIC GROUPS AND EISENSTEIN SERIES: AN INTRODUCTION, II
COHOMOLOGY OF ARITHMETIC GROUPS AND EISENSTEIN SERIES: AN INTRODUCTION, II LECTURES BY JOACHIM SCHWERMER, NOTES BY TONY FENG Contents 1. Review 1 2. Lifting differential forms from the boundary 2 3. Eisenstein
More informationBESSEL MODELS FOR GSp(4)
BESSEL MODELS FOR GSp(4) DIPENDRA PRASAD AND RAMIN TAKLOO-BIGHASH To Steve Gelbart Abstract. Methods of theta correspondence are used to analyze local and global Bessel models for GSp 4 proving a conjecture
More information0 A. ... A j GL nj (F q ), 1 j r
CHAPTER 4 Representations of finite groups of Lie type Let F q be a finite field of order q and characteristic p. Let G be a finite group of Lie type, that is, G is the F q -rational points of a connected
More informationA partition of the set of enhanced Langlands parameters of a reductive p-adic group
A partition of the set of enhanced Langlands parameters of a reductive p-adic group joint work with Ahmed Moussaoui and Maarten Solleveld Anne-Marie Aubert Institut de Mathématiques de Jussieu - Paris
More informationTHE GROSS PRASAD CONJECTURE AND LOCAL THETA CORRESPONDENCE. 1. Introduction
THE GROSS RASAD CONJECTURE AND LOCAL THETA CORRESONDENCE WEE TECK GAN AND ATSUSHI ICHINO Abstract. We establish the Fourier Jacobi case of the local Gross rasad conjecture for unitary groups, by using
More informationDirac Cohomology, Orbit Method and Unipotent Representations
Dirac Cohomology, Orbit Method and Unipotent Representations Dedicated to Bert Kostant with great admiration Jing-Song Huang, HKUST Kostant Conference MIT, May 28 June 1, 2018 coadjoint orbits of reductive
More informationThe Endoscopic Classification of Representations
The Endoscopic Classification of Representations James Arthur We shall outline a classification [A] of the automorphic representations of special orthogonal and symplectic groups in terms of those of general
More informationON THE LOCAL LANGLANDS CONJECTURE AND RELATED PROBLEMS OVER p-adic LOCAL FIELDS
ON THE LOCAL LANGLANDS CONJECTURE AND RELATED PROBLEMS OVER p-adic LOCAL FIELDS DIHUA JIANG AND CHUFENG NIEN Abstract. We discuss two issues related to the local Langlands conjecture over p-adic local
More informationOn Local γ-factors. Dihua Jiang School of Mathematics, University of Minnesota Minneapolis, MN 55455, USA. February 27, 2006.
On Local γ-factors Dihua Jiang School of Mathematics, University of Minnesota Minneapolis, MN 55455, USA February 27, 2006 Contents 1 Introduction 1 2 Basic Properties of Local γ-factors 5 2.1 Multiplicativity..........................
More informationColette Mœglin and Marko Tadić
CONSTRUCTION OF DISCRETE SERIES FOR CLASSICAL p-adic GROUPS Colette Mœglin and Marko Tadić Introduction The goal of this paper is to complete (after [M2] the classification of irreducible square integrable
More informationON RESIDUAL COHOMOLOGY CLASSES ATTACHED TO RELATIVE RANK ONE EISENSTEIN SERIES FOR THE SYMPLECTIC GROUP
ON RESIDUAL COHOMOLOGY CLASSES ATTACHED TO RELATIVE RANK ONE EISENSTEIN SERIES FOR THE SYMPLECTIC GROUP NEVEN GRBAC AND JOACHIM SCHWERMER Abstract. The cohomology of an arithmetically defined subgroup
More informationLIFTING OF GENERIC DEPTH ZERO REPRESENTATIONS OF CLASSICAL GROUPS
LIFTING OF GENERIC DEPTH ZERO REPRESENTATIONS OF CLASSICAL GROUPS GORDAN SAVIN 1. Introduction Let G be a split classical group, either SO 2n+1 (k) or Sp 2n (k), over a p-adic field k. We assume that p
More informationSYMMETRIC SUBGROUP ACTIONS ON ISOTROPIC GRASSMANNIANS
1 SYMMETRIC SUBGROUP ACTIONS ON ISOTROPIC GRASSMANNIANS HUAJUN HUANG AND HONGYU HE Abstract. Let G be the group preserving a nondegenerate sesquilinear form B on a vector space V, and H a symmetric subgroup
More informationA brief overview of modular and automorphic forms
A brief overview of modular and automorphic forms Kimball Martin Original version: Fall 200 Revised version: June 9, 206 These notes were originally written in Fall 200 to provide a very quick overview
More informationIrreducible subgroups of algebraic groups
Irreducible subgroups of algebraic groups Martin W. Liebeck Department of Mathematics Imperial College London SW7 2BZ England Donna M. Testerman Department of Mathematics University of Lausanne Switzerland
More informationOn exceptional completions of symmetric varieties
Journal of Lie Theory Volume 16 (2006) 39 46 c 2006 Heldermann Verlag On exceptional completions of symmetric varieties Rocco Chirivì and Andrea Maffei Communicated by E. B. Vinberg Abstract. Let G be
More informationBranching rules of unitary representations: Examples and applications to automorphic forms.
Branching rules of unitary representations: Examples and applications to automorphic forms. Basic Notions: Jerusalem, March 2010 Birgit Speh Cornell University 1 Let G be a group and V a vector space.
More informationNon-tempered Arthur Packets of G Introduction
Non-tempered Arthur Packets of G 2 Wee Teck Gan and Nadya Gurevich to Professor S. Rallis with admiration and best wishes 1. Introduction In his famous conjecture, J. Arthur gave a description of the discrete
More informationWhittaker models and Fourier coeffi cients of automorphic forms
Whittaker models and Fourier coeffi cients of automorphic forms Nolan R. Wallach May 2013 N. Wallach () Whittaker models 5/13 1 / 20 G be a real reductive group with compact center and let K be a maximal
More informationarxiv: v1 [math.rt] 14 Nov 2007
arxiv:0711.2128v1 [math.rt] 14 Nov 2007 SUPPORT VARIETIES OF NON-RESTRICTED MODULES OVER LIE ALGEBRAS OF REDUCTIVE GROUPS: CORRIGENDA AND ADDENDA ALEXANDER PREMET J. C. Jantzen informed me that the proof
More informationAUTOMORPHIC FORMS NOTES, PART I
AUTOMORPHIC FORMS NOTES, PART I DANIEL LITT The goal of these notes are to take the classical theory of modular/automorphic forms on the upper half plane and reinterpret them, first in terms L 2 (Γ \ SL(2,
More informationARCHIMEDEAN ASPECTS OF SIEGEL MODULAR FORMS OF DEGREE 2
ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 47, Number 7, 207 ARCHIMEDEAN ASPECTS OF SIEGEL MODULAR FORMS OF DEGREE 2 RALF SCHMIDT ABSTRACT. We survey the archimedean representations and Langlands parameters
More informationPeriods of Automorpic Forms
Periods of Automorpic Forms Dihua Jiang Abstract. In this paper, we discuss periods of automorphic forms from the representation-theoretic point of view. It gives general theory of periods and some special
More informationFunctoriality and the trace formula
Functoriality and the trace formula organized by Ali Altug, James Arthur, Bill Casselman, and Tasho Kaletha Workshop Summary This workshop was devoted to exploring the future of Langlands functoriality
More informationThe Grothendieck-Katz Conjecture for certain locally symmetric varieties
The Grothendieck-Katz Conjecture for certain locally symmetric varieties Benson Farb and Mark Kisin August 20, 2008 Abstract Using Margulis results on lattices in semi-simple Lie groups, we prove the Grothendieck-
More informationFundamental Lemma and Hitchin Fibration
Fundamental Lemma and Hitchin Fibration Gérard Laumon CNRS and Université Paris-Sud May 13, 2009 Introduction In this talk I shall mainly report on Ngô Bao Châu s proof of the Langlands-Shelstad Fundamental
More informationRepresentation Theory and Orbital Varieties. Thomas Pietraho Bowdoin College
Representation Theory and Orbital Varieties Thomas Pietraho Bowdoin College 1 Unitary Dual Problem Definition. A representation (π, V ) of a group G is a homomorphism ρ from G to GL(V ), the set of invertible
More informationSTABILITY AND ENDOSCOPY: INFORMAL MOTIVATION. James Arthur University of Toronto
STABILITY AND ENDOSCOPY: INFORMAL MOTIVATION James Arthur University of Toronto The purpose of this note is described in the title. It is an elementary introduction to some of the basic ideas of stability
More informationRIMS L. Title: Abstract:,,
& 2 1 ( ) RIMS L 13:30 14:30 ( ) Title: Whittaker functions on Sp(2,R) and archimedean zeta integrals. There are 4 kinds of generic representations of Sp(2,R), and explicit formulas of Whittaker functions
More informationRIMS. Ibukiyama Zhuravlev. B.Heim
RIMS ( ) 13:30-14:30 ( ) Title: Generalized Maass relations and lifts. Abstract: (1) Duke-Imamoglu-Ikeda Eichler-Zagier- Ibukiyama Zhuravlev L- L- (2) L- L- L B.Heim 14:45-15:45 ( ) Title: Kaneko-Zagier
More informationThe Contragredient. Spherical Unitary Dual for Complex Classical Groups
The Contragredient Joint with D. Vogan Spherical Unitary Dual for Complex Classical Groups Joint with D. Barbasch The Contragredient Problem: Compute the involution φ φ of the space of L-homomorphisms
More informationFourier coefficients and nilpotent orbits for small representations
Fourier coefficients and nilpotent orbits for small representations Axel Kleinschmidt (Albert Einstein Institute, Potsdam) Workshop on Eisenstein series on Kac Moody groups KIAS, Seoul, Nov 18, 2015 With
More informationDiscrete Series and Characters of the Component Group
Discrete Series and Characters of the Component Group Jeffrey Adams April 9, 2007 Suppose φ : W R L G is an L-homomorphism. There is a close relationship between the L-packet associated to φ and characters
More informationAutomorphic Galois representations and Langlands correspondences
Automorphic Galois representations and Langlands correspondences II. Attaching Galois representations to automorphic forms, and vice versa: recent progress Bowen Lectures, Berkeley, February 2017 Outline
More informationDUALITY, CENTRAL CHARACTERS, AND REAL-VALUED CHARACTERS OF FINITE GROUPS OF LIE TYPE
DUALITY, CENTRAL CHARACTERS, AND REAL-VALUED CHARACTERS OF FINITE GROUPS OF LIE TYPE C. RYAN VINROOT Abstract. We prove that the duality operator preserves the Frobenius- Schur indicators of characters
More information(E.-W. Zink, with A. Silberger)
1 Langlands classification for L-parameters A talk dedicated to Sergei Vladimirovich Vostokov on the occasion of his 70th birthday St.Petersburg im Mai 2015 (E.-W. Zink, with A. Silberger) In the representation
More informationTempered endoscopy for real groups III: inversion of transfer and L-packet structure
Tempered endoscopy for real groups III: inversion of transfer and L-packet structure D. Shelstad 1. Introduction This is the last part of a project in which we use the canonical (geometric) transfer factors
More informationMath 210C. The representation ring
Math 210C. The representation ring 1. Introduction Let G be a nontrivial connected compact Lie group that is semisimple and simply connected (e.g., SU(n) for n 2, Sp(n) for n 1, or Spin(n) for n 3). Let
More informationUnipotent Representations and the Dual Pairs Correspondence
Unipotent Representations and the Dual Pairs Correspondence Dan Barbasch Yale June 015 August 7, 015 1 / 35 Introduction I first met Roger Howe at a conference in Luminy in 1978. At the time I knew some
More informationREDUCIBLE PRINCIPAL SERIES REPRESENTATIONS, AND LANGLANDS PARAMETERS FOR REAL GROUPS. May 15, 2018 arxiv: v2 [math.
REDUCIBLE PRINCIPAL SERIES REPRESENTATIONS, AND LANGLANDS PARAMETERS FOR REAL GROUPS DIPENDRA PRASAD May 15, 2018 arxiv:1705.01445v2 [math.rt] 14 May 2018 Abstract. The work of Bernstein-Zelevinsky and
More informationFOURIER COEFFICIENTS AND SMALL AUTOMORPHIC REPRESENTATIONS
FOURIER COEFFICIENTS AND SMALL AUTOMORPHIC REPRESENTATIONS DMITRY GOUREVITCH, HENRIK P. A. GUSTAFSSON, AXEL KLEINSCHMIDT, DANIEL PERSSON, AND SIDDHARTHA SAHI arxiv:1811.05966v1 [math.nt] 14 Nov 2018 Abstract.
More informationHecke modifications. Aron Heleodoro. May 28, 2013
Hecke modifications Aron Heleodoro May 28, 2013 1 Introduction The interest on Hecke modifications in the geometrical Langlands program comes as a natural categorification of the product in the spherical
More informationUnitarity of non-spherical principal series
Unitarity of non-spherical principal series Alessandra Pantano July 2005 1 Minimal Principal Series G: a real split semisimple Lie group θ: Cartan involution; g = k p: Cartan decomposition of g a: maximal
More informationLecture 4: LS Cells, Twisted Induction, and Duality
Lecture 4: LS Cells, Twisted Induction, and Duality B. Binegar Department of Mathematics Oklahoma State University Stillwater, OK 74078, USA Nankai Summer School in Representation Theory and Harmonic Analysis
More informationFUNCTORIALITY FOR THE CLASSICAL GROUPS
FUNCTORIALITY FOR THE CLASSICAL GROUPS by J.W. COGDELL, H.H. KIM, I.I. PIATETSKI-SHAPIRO and F. SHAHIDI Functoriality is one of the most central questions in the theory of automorphic forms and representations
More informationarxiv: v1 [math.rt] 22 Mar 2015
ON CERTAIN GLOBAL CONSTRUCTIONS OF AUTOMORPHIC FORMS RELATED TO SMALL REPRESENTATIONS OF F 4 DAVID GINZBURG arxiv:1503.06409v1 [math.rt] 22 Mar 2015 To the memory of S. Rallis Abstract. In this paper we
More informationNotes on nilpotent orbits Computational Theory of Real Reductive Groups Workshop. Eric Sommers
Notes on nilpotent orbits Computational Theory of Real Reductive Groups Workshop Eric Sommers 17 July 2009 2 Contents 1 Background 5 1.1 Linear algebra......................................... 5 1.1.1
More informationOn the Self-dual Representations of a p-adic Group
IMRN International Mathematics Research Notices 1999, No. 8 On the Self-dual Representations of a p-adic Group Dipendra Prasad In an earlier paper [P1], we studied self-dual complex representations of
More informationCUSPIDALITY OF SYMMETRIC POWERS WITH APPLICATIONS
DUKE MATHEMATICAL JOURNAL Vol. 112, No. 1, c 2002 CUSPIDALITY OF SYMMETRIC POWERS WITH APPLICATIONS HENRY H. KIM and FREYDOON SHAHIDI Abstract The purpose of this paper is to prove that the symmetric fourth
More informationSOME PROPERTIES OF CHARACTER SHEAVES. Anne-Marie Aubert. Dedicated to the memory of Olga Taussky-Todd. 1. Introduction.
pacific journal of mathematics Vol. 181, No. 3, 1997 SOME PROPERTIES OF CHARACTER SHEAVES Anne-Marie Aubert Dedicated to the memory of Olga Taussky-Todd 1. Introduction. In 1986, George Lusztig stated
More informationOn the Irreducibility of the Commuting Variety of the Symmetric Pair so p+2, so p so 2
Journal of Lie Theory Volume 16 (2006) 57 65 c 2006 Heldermann Verlag On the Irreducibility of the Commuting Variety of the Symmetric Pair so p+2, so p so 2 Hervé Sabourin and Rupert W.T. Yu Communicated
More informationVARIATIONS ON THE BAER SUZUKI THEOREM. 1. Introduction
VARIATIONS ON THE BAER SUZUKI THEOREM ROBERT GURALNICK AND GUNTER MALLE Dedicated to Bernd Fischer on the occasion of his 75th birthday Abstract. The Baer Suzuki theorem says that if p is a prime, x is
More informationEKT of Some Wonderful Compactifications
EKT of Some Wonderful Compactifications and recent results on Complete Quadrics. (Based on joint works with Soumya Banerjee and Michael Joyce) Mahir Bilen Can April 16, 2016 Mahir Bilen Can EKT of Some
More informationON BASE SIZES FOR ALGEBRAIC GROUPS
ON BASE SIZES FOR ALGEBRAIC GROUPS TIMOTHY C. BURNESS, ROBERT M. GURALNICK, AND JAN SAXL Abstract. For an algebraic group G and a closed subgroup H, the base size of G on the coset variety of H in G is
More informationSPHERICAL UNITARY DUAL FOR COMPLEX CLASSICAL GROUPS
October 3, 008 SPHERICAL UNITARY DUAL FOR COMPLEX CLASSICAL GROUPS DAN BARBASCH 1. Introduction The full unitary dual for the complex classical groups viewed as real Lie groups is computed in [B1]. This
More informationThe Langlands Classification and Irreducible Characters for Real Reductive Groups JEFFREY ADAMS DAN BARBASCH DAVID A. VOGAN, JR.
The Langlands Classification and Irreducible Characters for Real Reductive Groups JEFFREY ADAMS DAN BARBASCH DAVID A. VOGAN, JR. 1. Introduction. In [2] and [3], Arthur has formulated a number of conjectures
More informationToshiaki Shoji (Nagoya University) Character sheaves on a symmetric space and Kostka polynomials July 27, 2012, Osaka 1 / 1
Character sheaves on a symmetric space and Kostka polynomials Toshiaki Shoji Nagoya University July 27, 2012, Osaka Character sheaves on a symmetric space and Kostka polynomials July 27, 2012, Osaka 1
More informationON SOME PARTITIONS OF A FLAG MANIFOLD. G. Lusztig
ON SOME PARTITIONS OF A FLAG MANIFOLD G. Lusztig Introduction Let G be a connected reductive group over an algebraically closed field k of characteristic p 0. Let W be the Weyl group of G. Let W be the
More informationVANISHING OF CERTAIN EQUIVARIANT DISTRIBUTIONS ON SPHERICAL SPACES
VANISHING OF CERTAIN EQUIVARIANT DISTRIBUTIONS ON SPHERICAL SPACES AVRAHAM AIZENBUD AND DMITRY GOUREVITCH Abstract. We prove vanishing of z-eigen distributions on a spherical variety of a split real reductive
More informationCounting matrices over finite fields
Counting matrices over finite fields Steven Sam Massachusetts Institute of Technology September 30, 2011 1/19 Invertible matrices F q is a finite field with q = p r elements. [n] = 1 qn 1 q = qn 1 +q n
More informationMath 249B. Geometric Bruhat decomposition
Math 249B. Geometric Bruhat decomposition 1. Introduction Let (G, T ) be a split connected reductive group over a field k, and Φ = Φ(G, T ). Fix a positive system of roots Φ Φ, and let B be the unique
More informationGelfand Pairs and Invariant Distributions
Gelfand Pairs and Invariant Distributions A. Aizenbud Massachusetts Institute of Technology http://math.mit.edu/~aizenr Examples Example (Fourier Series) Examples Example (Fourier Series) L 2 (S 1 ) =
More informationTHE RESIDUAL EISENSTEIN COHOMOLOGY OF Sp 4 OVER A TOTALLY REAL NUMBER FIELD
THE RESIDUAL EISENSTEIN COHOMOLOGY OF Sp 4 OVER A TOTALLY REAL NUMBER FIELD NEVEN GRBAC AND HARALD GROBNER Abstract. Let G = Sp 4 /k be the k-split symplectic group of k-rank, where k is a totally real
More informationKIRILLOV THEORY AND ITS APPLICATIONS
KIRILLOV THEORY AND ITS APPLICATIONS LECTURE BY JU-LEE KIM, NOTES BY TONY FENG Contents 1. Motivation (Akshay Venkatesh) 1 2. Howe s Kirillov theory 2 3. Moy-Prasad theory 5 4. Applications 6 References
More informationSubquotients of Minimal Principal Series
Subquotients of Minimal Principal Series ν 1 = ν 2 Sp(4) 0 1 2 ν 2 = 0 Alessandra Pantano, UCI July 2008 1 PART 1 Introduction Preliminary Definitions and Notation 2 Langlands Quotients of Minimal Principal
More informationOn Partial Poincaré Series
Contemporary Mathematics On Partial Poincaré Series J.W. Cogdell and I.I. Piatetski-Shapiro This paper is dedicated to our colleague and friend Steve Gelbart. Abstract. The theory of Poincaré series has
More informationA REMARK ON A CONVERSE THEOREM OF COGDELL AND PIATETSKI-SHAPIRO
A REMARK ON A CONVERSE THEOREM OF COGDELL AND PIATETSKI-SHAPIRO HERVÉ JACQUET AND BAIYING LIU Abstract. In this paper, we reprove a global converse theorem of Cogdell and Piatetski-Shapiro using purely
More information260 I.I. PIATETSKI-SHAPIRO and one can associate to f() a Dirichlet series L(f; s) = X T modulo integral equivalence a T jt j s : Hecke's original pro
pacific journal of mathematics Vol. 181, No. 3, 1997 L-FUNCTIONS FOR GSp 4 I.I. Piatetski-Shapiro Dedicated to Olga Taussky-Todd 1. Introduction. To a classical modular cusp form f(z) with Fourier expansion
More informationENDOSCOPY AND COHOMOLOGY OF U(n, 1)
ENDOSCOPY AND COHOMOLOGY OF U(n, 1) SIMON MARSHALL AND SUG WOO SHIN Abstract By assuming the endoscopic classification of automorphic representations on inner forms of unitary groups, which is currently
More informationParabolic subgroups Montreal-Toronto 2018
Parabolic subgroups Montreal-Toronto 2018 Alice Pozzi January 13, 2018 Alice Pozzi Parabolic subgroups Montreal-Toronto 2018 January 13, 2018 1 / 1 Overview Alice Pozzi Parabolic subgroups Montreal-Toronto
More informationHolomorphy of the 9th Symmetric Power L-Functions for Re(s) >1. Henry H. Kim and Freydoon Shahidi
IMRN International Mathematics Research Notices Volume 2006, Article ID 59326, Pages 1 7 Holomorphy of the 9th Symmetric Power L-Functions for Res >1 Henry H. Kim and Freydoon Shahidi We proe the holomorphy
More informationEigenvalue problem for Hermitian matrices and its generalization to arbitrary reductive groups
Eigenvalue problem for Hermitian matrices and its generalization to arbitrary reductive groups Shrawan Kumar Talk given at AMS Sectional meeting held at Davidson College, March 2007 1 Hermitian eigenvalue
More informationASYMPTOTICS OF THE NUMBER OF INVOLUTIONS IN FINITE CLASSICAL GROUPS
ASYMPTOTICS OF THE NUMBER OF INVOLUTIONS IN FINITE CLASSICAL GROUPS JASON FULMAN, ROBERT GURALNICK, AND DENNIS STANTON Abstract Answering a question of Geoff Robinson, we compute the large n iting proportion
More informationADELIC VERSION OF MARGULIS ARITHMETICITY THEOREM. Hee Oh
ADELIC VERSION OF MARGULIS ARITHMETICITY THEOREM Hee Oh Abstract. In this paper, we generalize Margulis s S-arithmeticity theorem to the case when S can be taken as an infinite set of primes. Let R be
More information