Regular Bernstein blocks

For a connected reductive group G defined over a non-archimedean local field F, we consider the Bernstein blocks in the category of smooth representations of G ⁢ ( F ) {G(F)} . Bernstein blocks whose cuspidal support involves a regular supercuspidal representation are called regular Bernstein blocks. Most Bernstein blocks are regular when the residual characteristic of F is not too small. Under mild hypotheses on the residual characteristic, we show that the Bernstein center of a regular Bernstein block of G ⁢ ( F ) {G(F)} is isomorphic to the Bernstein center of a regular depth-zero Bernstein block of G 0 ⁢ ( F ) {G^{0}(F)} , where G 0 {G^{0}} is a certain twisted Levi subgroup of G. In some cases, we show that the blocks themselves are equivalent, and as a consequence we prove the ABPS Conjecture in some new cases.

Rodier type theorem for generalized principal series

Given a regular supercuspidal representation $$\rho $$ ρ of the Levi subgroup M of a standard parabolic subgroup $$P=MN$$ P = M N in a connected reductive group G defined over a non-archimedean local field F, we serve you a Rodier type structure theorem which provides us a geometrical parametrization of the set $$JH(Ind^G_P(\rho ))$$ J H ( I n d P G ( ρ ) ) of Jordan–Hölder constituents of the Harish-Chandra parabolic induction representation $$Ind^G_P(\rho )$$ I n d P G ( ρ ) , vastly generalizing Rodier structure theorem for $$P=B=TU$$ P = B = T U Borel subgroup of a connected split reductive group about 40 years ago. Our novel contribution is to overcome the essential difficulty that the relative Weyl group $$W_M=N_G(M)/M$$ W M = N G ( M ) / M is not a coxeter group in general, as opposed to the well-known fact that the Weyl group $$W_T=N_G(T)/T$$ W T = N G ( T ) / T is a coxeter group. Along the way, we sort out all regular discrete series/tempered/generic representations for arbitrary G, generalizing Tadić’s work on regular discrete series representation for split $$(G)Sp_{2n}$$ ( G ) S p 2 n and $$SO_{2n+1}$$ S O 2 n + 1 , and also providing a new simple proof of Casselman–Shahidi’s theorem on generalized injectivity conjecture for regular generalized principal series. Indeed, such a beautiful structure theorem also holds for finite central covering groups.

THE COHOMOLOGY OF UNRAMIFIED RAPOPORT–ZINK SPACES OF EL-TYPE AND HARRIS'S CONJECTURE

We study the l-adic cohomology of unramified Rapoport–Zink spaces of EL-type. These spaces were used in Harris and Taylor's proof of the local Langlands correspondence for $\mathrm {GL_n}$ and to show local–global compatibilities of the Langlands correspondence. In this paper we consider certain morphisms $\mathrm {Mant}_{b, \mu }$ of Grothendieck groups of representations constructed from the cohomology of these spaces, as studied by Harris and Taylor, Mantovan, Fargues, Shin and others. Due to earlier work of Fargues and Shin we have a description of $\mathrm {Mant}_{b, \mu }(\rho )$ for $\rho $ a supercuspidal representation. In this paper, we give a conjectural formula for $\mathrm {Mant}_{b, \mu }(\rho )$ for $\rho $ an admissible representation and prove it when $\rho $ is essentially square-integrable. Our proof works for general $\rho $ conditionally on a conjecture appearing in Shin's work. We show that our description agrees with a conjecture of Harris in the case of parabolic inductions of supercuspidal representations of a Levi subgroup.

Automorphic vector bundles on the stack of G-zips

For a connected reductive group G over a finite field, we study automorphic vector bundles on the stack of G-zips. In particular, we give a formula in the general case for the space of global sections of an automorphic vector bundle in terms of the Brylinski-Kostant filtration. Moreover, we give an equivalence of categories between the category of automorphic vector bundles on the stack of G-zips and a category of admissible modules with actions of a 0-dimensional algebraic subgroup a Levi subgroup and monodromy operators.

ON COMPACT HOMOGENEOUS -MANIFOLDS

We prove that among all compact homogeneous spaces for an effective transitive action of a Lie group whose Levi subgroup has no compact simple factors, the seven-dimensional flat torus is the only one that admits an invariant torsion-free $\text{G}_{2(2)}$ -structure.

Bernstein center of supercuspidal blocks

Let \mathbf{G} be a tamely ramified connected reductive group defined over a non-archimedean local field k. We show that the Bernstein center of a tame supercuspidal block of \mathbf{G}(k) is isomorphic to the Bernstein center of a depth-zero supercuspidal block of \mathbf{G}^{0}(k) for some twisted Levi subgroup of \mathbf{G}^{0} of \mathbf{G} .

Towards a Goldberg–Shahidi pairing for classical groups

Let{G^{1}}be an orthogonal, symplectic or unitary group over a local field and let{P=MN}be a maximal parabolic subgroup. Then the Levi subgroupMis the product of a group of the same type as{G^{1}}and a general linear group, acting on vector spacesXandW, respectively. In this paper we decompose the unipotent radicalNofPunder the adjoint action ofM, assuming{\dim W\leq\dim X}, excluding only the symplectic case with{\dim W}odd. The result is a Weyl-type integration formula forNwith applications to the theory of intertwining operators for parabolically induced representations of{G^{1}}. Namely, one obtains a bilinear pairing on matrix coefficients, in the spirit of Goldberg–Shahidi, which detects the presence of poles of these operators at 0.