We prove the local Langlands conjecture for the exceptional group $G_2(F)$ where F is a non-archi... more We prove the local Langlands conjecture for the exceptional group $G_2(F)$ where F is a non-archimedean local field of characteristic zero.
Let G be a general linear group over a p-adic field. It is well known that Bernstein components o... more Let G be a general linear group over a p-adic field. It is well known that Bernstein components of the category of smooth representations of G are described by Hecke algebras arising from Bushnell-Kutzko types. We describe the Bernstein components of the Gelfand-Graev representation of G by explicit Hecke algebra modules. This result is used to translate the theory of Bernstein-Zelevinsky derivatives in the language of representations of Hecke algebras, where we develop a comprehensive theory.
Let G be a split reductive group over a p-adic field F. Let B be a Borel subgroup and U the maxim... more Let G be a split reductive group over a p-adic field F. Let B be a Borel subgroup and U the maximal unipotent subgroup of B. Let ψ be a Whittaker character of U. Let I be an Iwahori subgroup of G. We describe the Iwahori-Hecke algebra action on the Gelfand-Graev representation (ind G U ψ) I by an explicit projective module. As a consequence, for G = GL(n, F), we define and describe Bernstein-Zelevinsky derivatives of representations generated by I-fixed vectors in terms of the corresponding Iwahori-Hecke algebra modules. Furthermore, using Lusztig's reductions, we show that the Bernstein-Zelevinsky derivatives can be determined using graded Hecke algebras. We give two applications of our study. Firstly, we compute the Bernstein-Zelevinsky derivatives of generalized Speh modules, which recovers a result of Lapid-Mínguez and Tadić. Secondly, we give a realization of the Iwahori-Hecke algebra action on some generic representations of GL(n + 1, F), restricted to GL(n, F), which is further used to verify a conjecture on an Ext-branching problem of D. Prasad for a class of examples.
In this paper, we define a notion of pseudo-spherical type for the two fold central extension of ... more In this paper, we define a notion of pseudo-spherical type for the two fold central extension of SL 2 (Q 2). We relate this definition to some results in classical modular forms of half integral weights.
We prove a conjecture of Dipendra Prasad on the Ext-branching problem from GLn+1(F) to GLn(F), wh... more We prove a conjecture of Dipendra Prasad on the Ext-branching problem from GLn+1(F) to GLn(F), where F is a p-adic field.
Let g = k ⊕ s be a Cartan decomposition of a simple complex Lie algebra corresponding to the Chev... more Let g = k ⊕ s be a Cartan decomposition of a simple complex Lie algebra corresponding to the Chevalley involution. It is well known that among the set of primitive ideals with the infinitesimal character 1 2 ρ, there is a unique maximal primitive ideal J. Let Q := U (g)/J. Let K be a connected compact subgroup with Lie algebra k so that the notion of (g, K)-modules is well defined. In this paper we show that Q K is isomorphic to U (k) K. In particular Q K is commutative. A consequence of this result is that if W is an irreducible (g, K)-module annihilated by J, then W is K-multiplicity free and two such irreducible (g, K)-modules with a common nonzero K-type are isomorphic.
In his reinterpretation of Gauss's composition law for binary quadratic forms, Bhargava determine... more In his reinterpretation of Gauss's composition law for binary quadratic forms, Bhargava determined the integral orbits of a prehomogeneous vector space which arises naturally in the structure theory of the split group Spin 8. We consider a twisted version of this prehomogeneous vector space which arises in quasisplit Spin E 8 , where E is an étale cubic algebra over a field F. We classify the generic orbits over F by twisted composition F-algebras of E-dimension 2.
Representation Theory of The American Mathematical Society, Jan 13, 2005
This paper gives a self-contained exposition of minimal representations. We introduce a notion of... more This paper gives a self-contained exposition of minimal representations. We introduce a notion of weakly minimal representations and prove a global rigidity result for them. We address issues of uniqueness and existence and prove many key properties of minimal representations needed for global applications.
Exceptional groups of type E6 contain dual pairs where one member is Spin(8), and the other is T ... more Exceptional groups of type E6 contain dual pairs where one member is Spin(8), and the other is T ⋊ Z/2Z, where T is a two-dimensional torus and the non-trivial element in Z/2Z acts on T by the inverse involution. We describe the correspondence of representations arising by restricting the minimal representation.
We study three exceptional theta correspondences for p-adic groups, where one member of the dual ... more We study three exceptional theta correspondences for p-adic groups, where one member of the dual pair is the exceptional group G2. We prove the Howe duality conjecture for these dual pairs and a dichotomy theorem, and determine explicitly the theta lifts of all non-cuspidal representations.
Fix a quadratic order over the ring of integers. An embedding of the quadratic order into a quate... more Fix a quadratic order over the ring of integers. An embedding of the quadratic order into a quaternionic order naturally gives an integral binary hermitian form over the quadratic order. We show that, in certain cases, this correspondence is a discriminant preserving bijection between the isomorphism classes of embeddings and integral binary hermitian forms.
In this paper we study compact dual pair correspondences arising from smallest representations of... more In this paper we study compact dual pair correspondences arising from smallest representations of non-linear covers of odd orthogonal groups. We identify representations appearing in these correspondences with subquotients of cohomologically induced representations.
Let g be a simple Lie algebra of type F 4 or E n defined over a local or global field k of charac... more Let g be a simple Lie algebra of type F 4 or E n defined over a local or global field k of characteristic zero. We show that g can be obtained by the Tits construction from an octonion algebra O and a cubic Jordan algebra J. In particular, g contains a dual pair h defined over k which is the direct sum of the derivation algebras of O and J. We determine the conjugacy classes of k-forms of h in g.
Representation Theory of The American Mathematical Society, Oct 11, 2012
The metaplectic group is defined by its oscillator or Weil representation. Using the types of the... more The metaplectic group is defined by its oscillator or Weil representation. Using the types of the Weil representations we define two Hecke algebras that govern two Bernstein's components containing the even and the odd Weil representation, respectively.
This article was published in an Elsevier journal. The attached copy is furnished to the author f... more This article was published in an Elsevier journal. The attached copy is furnished to the author for non-commercial research and education use, including for instruction at the author's institution, sharing with colleagues and providing to institution administration. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier's archiving and manuscript policies are encouraged to visit: http://www.elsevier.com/copyright
We prove Howe duality for the theta correspondence arising from the p-adic dual pair G 2 × (PU 3 ... more We prove Howe duality for the theta correspondence arising from the p-adic dual pair G 2 × (PU 3 ⋊ Z/2Z) inside the quasi-split group of type E 6 .
Proceedings of the American Mathematical Society, 2019
We consider affine buildings with refined chamber structure. For each vertex x x we construct a c... more We consider affine buildings with refined chamber structure. For each vertex x x we construct a contraction, based at x x , that is used to prove exactness of Schneider-Stuhler resolutions of arbitrary depth.
We prove the local Langlands conjecture for the exceptional group $G_2(F)$ where F is a non-archi... more We prove the local Langlands conjecture for the exceptional group $G_2(F)$ where F is a non-archimedean local field of characteristic zero.
Let G be a general linear group over a p-adic field. It is well known that Bernstein components o... more Let G be a general linear group over a p-adic field. It is well known that Bernstein components of the category of smooth representations of G are described by Hecke algebras arising from Bushnell-Kutzko types. We describe the Bernstein components of the Gelfand-Graev representation of G by explicit Hecke algebra modules. This result is used to translate the theory of Bernstein-Zelevinsky derivatives in the language of representations of Hecke algebras, where we develop a comprehensive theory.
Let G be a split reductive group over a p-adic field F. Let B be a Borel subgroup and U the maxim... more Let G be a split reductive group over a p-adic field F. Let B be a Borel subgroup and U the maximal unipotent subgroup of B. Let ψ be a Whittaker character of U. Let I be an Iwahori subgroup of G. We describe the Iwahori-Hecke algebra action on the Gelfand-Graev representation (ind G U ψ) I by an explicit projective module. As a consequence, for G = GL(n, F), we define and describe Bernstein-Zelevinsky derivatives of representations generated by I-fixed vectors in terms of the corresponding Iwahori-Hecke algebra modules. Furthermore, using Lusztig's reductions, we show that the Bernstein-Zelevinsky derivatives can be determined using graded Hecke algebras. We give two applications of our study. Firstly, we compute the Bernstein-Zelevinsky derivatives of generalized Speh modules, which recovers a result of Lapid-Mínguez and Tadić. Secondly, we give a realization of the Iwahori-Hecke algebra action on some generic representations of GL(n + 1, F), restricted to GL(n, F), which is further used to verify a conjecture on an Ext-branching problem of D. Prasad for a class of examples.
In this paper, we define a notion of pseudo-spherical type for the two fold central extension of ... more In this paper, we define a notion of pseudo-spherical type for the two fold central extension of SL 2 (Q 2). We relate this definition to some results in classical modular forms of half integral weights.
We prove a conjecture of Dipendra Prasad on the Ext-branching problem from GLn+1(F) to GLn(F), wh... more We prove a conjecture of Dipendra Prasad on the Ext-branching problem from GLn+1(F) to GLn(F), where F is a p-adic field.
Let g = k ⊕ s be a Cartan decomposition of a simple complex Lie algebra corresponding to the Chev... more Let g = k ⊕ s be a Cartan decomposition of a simple complex Lie algebra corresponding to the Chevalley involution. It is well known that among the set of primitive ideals with the infinitesimal character 1 2 ρ, there is a unique maximal primitive ideal J. Let Q := U (g)/J. Let K be a connected compact subgroup with Lie algebra k so that the notion of (g, K)-modules is well defined. In this paper we show that Q K is isomorphic to U (k) K. In particular Q K is commutative. A consequence of this result is that if W is an irreducible (g, K)-module annihilated by J, then W is K-multiplicity free and two such irreducible (g, K)-modules with a common nonzero K-type are isomorphic.
In his reinterpretation of Gauss's composition law for binary quadratic forms, Bhargava determine... more In his reinterpretation of Gauss's composition law for binary quadratic forms, Bhargava determined the integral orbits of a prehomogeneous vector space which arises naturally in the structure theory of the split group Spin 8. We consider a twisted version of this prehomogeneous vector space which arises in quasisplit Spin E 8 , where E is an étale cubic algebra over a field F. We classify the generic orbits over F by twisted composition F-algebras of E-dimension 2.
Representation Theory of The American Mathematical Society, Jan 13, 2005
This paper gives a self-contained exposition of minimal representations. We introduce a notion of... more This paper gives a self-contained exposition of minimal representations. We introduce a notion of weakly minimal representations and prove a global rigidity result for them. We address issues of uniqueness and existence and prove many key properties of minimal representations needed for global applications.
Exceptional groups of type E6 contain dual pairs where one member is Spin(8), and the other is T ... more Exceptional groups of type E6 contain dual pairs where one member is Spin(8), and the other is T ⋊ Z/2Z, where T is a two-dimensional torus and the non-trivial element in Z/2Z acts on T by the inverse involution. We describe the correspondence of representations arising by restricting the minimal representation.
We study three exceptional theta correspondences for p-adic groups, where one member of the dual ... more We study three exceptional theta correspondences for p-adic groups, where one member of the dual pair is the exceptional group G2. We prove the Howe duality conjecture for these dual pairs and a dichotomy theorem, and determine explicitly the theta lifts of all non-cuspidal representations.
Fix a quadratic order over the ring of integers. An embedding of the quadratic order into a quate... more Fix a quadratic order over the ring of integers. An embedding of the quadratic order into a quaternionic order naturally gives an integral binary hermitian form over the quadratic order. We show that, in certain cases, this correspondence is a discriminant preserving bijection between the isomorphism classes of embeddings and integral binary hermitian forms.
In this paper we study compact dual pair correspondences arising from smallest representations of... more In this paper we study compact dual pair correspondences arising from smallest representations of non-linear covers of odd orthogonal groups. We identify representations appearing in these correspondences with subquotients of cohomologically induced representations.
Let g be a simple Lie algebra of type F 4 or E n defined over a local or global field k of charac... more Let g be a simple Lie algebra of type F 4 or E n defined over a local or global field k of characteristic zero. We show that g can be obtained by the Tits construction from an octonion algebra O and a cubic Jordan algebra J. In particular, g contains a dual pair h defined over k which is the direct sum of the derivation algebras of O and J. We determine the conjugacy classes of k-forms of h in g.
Representation Theory of The American Mathematical Society, Oct 11, 2012
The metaplectic group is defined by its oscillator or Weil representation. Using the types of the... more The metaplectic group is defined by its oscillator or Weil representation. Using the types of the Weil representations we define two Hecke algebras that govern two Bernstein's components containing the even and the odd Weil representation, respectively.
This article was published in an Elsevier journal. The attached copy is furnished to the author f... more This article was published in an Elsevier journal. The attached copy is furnished to the author for non-commercial research and education use, including for instruction at the author's institution, sharing with colleagues and providing to institution administration. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier's archiving and manuscript policies are encouraged to visit: http://www.elsevier.com/copyright
We prove Howe duality for the theta correspondence arising from the p-adic dual pair G 2 × (PU 3 ... more We prove Howe duality for the theta correspondence arising from the p-adic dual pair G 2 × (PU 3 ⋊ Z/2Z) inside the quasi-split group of type E 6 .
Proceedings of the American Mathematical Society, 2019
We consider affine buildings with refined chamber structure. For each vertex x x we construct a c... more We consider affine buildings with refined chamber structure. For each vertex x x we construct a contraction, based at x x , that is used to prove exactness of Schneider-Stuhler resolutions of arbitrary depth.
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