This paper studies homological properties of irreducible representations restricted from GL_n+1(F... more This paper studies homological properties of irreducible representations restricted from GL_n+1(F) to GL_n(F). We establish the following: (1) classify irreducible smooth representations of GL_n+1(F) which are projective when restricted to GL_n(F); (2) prove that each Bernstein component of an irreducible smooth representations of GL_n+1(F), restricted to GL_n(F), is indecomposable. In appendixes, we study some aspects of Speh representations, and in particular we give an explicit formulae of Ext-groups between Speh representations in terms of symmetric group representations and discuss the related Ext-brancing law.
Let $G$ be a split reductive group over a $p$-adic field $F$. Let $B$ be a Borel subgroup and $U$... 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 $\psi$ 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 $(\mathrm{ind}_{U}^{G}\psi)^I$ 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-M\'inguez and Tadi\'c. Secondly, we give a realization of the Iwahori-Hecke algebra action on some generic representation...
Journal für die reine und angewandte Mathematik (Crelles Journal), 2021
We prove a local Gan–Gross–Prasad conjecture on predicting the branching law for the non-tempered... more We prove a local Gan–Gross–Prasad conjecture on predicting the branching law for the non-tempered representations of general linear groups in the case of non-Archimedean fields. We also generalize to Bessel and Fourier–Jacobi models and study a possible generalization to Ext-branching laws.
In this paper, we establish connections between the first extensions of simple modules and certai... more In this paper, we establish connections between the first extensions of simple modules and certain filtrations of of standard modules in the setting of graded Hecke algebras. The filtrations involved are radical filtrations and Jantzen filtrations. Our approach involves the use of information from the Langlands classification as well as some deeper analysis on structure of some modules. Such modules arise from the image of a Knapp-Stein type intertwining operator and is a quotient of a generalized standard module.
LetFbe a non-Archimedean local field. This paper studies homological properties of irreducible sm... more LetFbe a non-Archimedean local field. This paper studies homological properties of irreducible smooth representations restricted from$${\mathrm {GL}}_{n+1}(F)$$GLn+1(F)to$${\mathrm {GL}}_n(F)$$GLn(F). A main result shows that each Bernstein component of an irreducible smooth representation of$${\mathrm {GL}}_{n+1}(F)$$GLn+1(F)restricted to$${\mathrm {GL}}_n(F)$$GLn(F)is indecomposable. We also classify all irreducible representations which are projective when restricting from$${\mathrm {GL}}_{n+1}(F)$$GLn+1(F)to$${\mathrm {GL}}_n(F)$$GLn(F). A main tool of our study is a notion of left and right derivatives, extending some previous work joint with Gordan Savin. As a by-product, we also determine the branching law in the opposite direction.
Abstract. For a connected complex semi-simple Lie group G and a fixed pair (B,B−) of opposite Bor... more Abstract. For a connected complex semi-simple Lie group G and a fixed pair (B,B−) of opposite Borel subgroups of G, we determine when the intersection of a conjugacy class C in G and a double coset BwB − is non-empty, where w is in the Weyl group W of G. The question comes from Poisson geometry, and our answer is in terms of the Bruhat order on W and an involution mC ∈ W associated to C. We study properties of the elements mC. For G = SL(n+1,C), we describe mC explicitly for every conjugacy class C, and for the case when w ∈ W is an involution, we also give an explicit answer to when C ∩ (BwB) is non-empty. 1.
This paper studies homological properties of irreducible representations restricted from GL_n+1(F... more This paper studies homological properties of irreducible representations restricted from GL_n+1(F) to GL_n(F). We establish the following: (1) classify irreducible smooth representations of GL_n+1(F) which are projective when restricted to GL_n(F); (2) prove that each Bernstein component of an irreducible smooth representations of GL_n+1(F), restricted to GL_n(F), is indecomposable. In appendixes, we study some aspects of Speh representations, and in particular we give an explicit formulae of Ext-groups between Speh representations in terms of symmetric group representations and discuss the related Ext-brancing law.
Let $G$ be a split reductive group over a $p$-adic field $F$. Let $B$ be a Borel subgroup and $U$... 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 $\psi$ 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 $(\mathrm{ind}_{U}^{G}\psi)^I$ 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-M\'inguez and Tadi\'c. Secondly, we give a realization of the Iwahori-Hecke algebra action on some generic representation...
Journal für die reine und angewandte Mathematik (Crelles Journal), 2021
We prove a local Gan–Gross–Prasad conjecture on predicting the branching law for the non-tempered... more We prove a local Gan–Gross–Prasad conjecture on predicting the branching law for the non-tempered representations of general linear groups in the case of non-Archimedean fields. We also generalize to Bessel and Fourier–Jacobi models and study a possible generalization to Ext-branching laws.
In this paper, we establish connections between the first extensions of simple modules and certai... more In this paper, we establish connections between the first extensions of simple modules and certain filtrations of of standard modules in the setting of graded Hecke algebras. The filtrations involved are radical filtrations and Jantzen filtrations. Our approach involves the use of information from the Langlands classification as well as some deeper analysis on structure of some modules. Such modules arise from the image of a Knapp-Stein type intertwining operator and is a quotient of a generalized standard module.
LetFbe a non-Archimedean local field. This paper studies homological properties of irreducible sm... more LetFbe a non-Archimedean local field. This paper studies homological properties of irreducible smooth representations restricted from$${\mathrm {GL}}_{n+1}(F)$$GLn+1(F)to$${\mathrm {GL}}_n(F)$$GLn(F). A main result shows that each Bernstein component of an irreducible smooth representation of$${\mathrm {GL}}_{n+1}(F)$$GLn+1(F)restricted to$${\mathrm {GL}}_n(F)$$GLn(F)is indecomposable. We also classify all irreducible representations which are projective when restricting from$${\mathrm {GL}}_{n+1}(F)$$GLn+1(F)to$${\mathrm {GL}}_n(F)$$GLn(F). A main tool of our study is a notion of left and right derivatives, extending some previous work joint with Gordan Savin. As a by-product, we also determine the branching law in the opposite direction.
Abstract. For a connected complex semi-simple Lie group G and a fixed pair (B,B−) of opposite Bor... more Abstract. For a connected complex semi-simple Lie group G and a fixed pair (B,B−) of opposite Borel subgroups of G, we determine when the intersection of a conjugacy class C in G and a double coset BwB − is non-empty, where w is in the Weyl group W of G. The question comes from Poisson geometry, and our answer is in terms of the Bruhat order on W and an involution mC ∈ W associated to C. We study properties of the elements mC. For G = SL(n+1,C), we describe mC explicitly for every conjugacy class C, and for the case when w ∈ W is an involution, we also give an explicit answer to when C ∩ (BwB) is non-empty. 1.
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