Base change for ramified unitary groups: The strongly ramified case

We compute a special case of base change of certain supercuspidal representations from a ramified unitary group to a general linear group, both defined over a p-adic field of odd residual characteristic. In this special case, we require the given supercuspidal representation to contain a skew maximal simple stratum, and the field datum of this stratum to be of maximal degree, tamely ramified over the base field, and quadratic ramified over its subfield fixed by the Galois involution that defines the unitary group. The base change of this supercuspidal representation is described by a canonical lifting of its underlying simple character, together with the base change of the level-zero component of its inducing cuspidal type, modified by a sign attached to a quadratic Gauss sum defined by the internal structure of the simple character. To obtain this result, we study the reducibility points of a parabolic induction and the corresponding module over the affine Hecke algebra, defined by the covering type over the product of types of the given supercuspidal representation and of a candidate of its base change.

A Characteristic Map for the Symmetric Space of Symplectic Forms Over a Finite Field

The characteristic map for the symmetric group is an isomorphism relating the representation theory of the symmetric group to symmetric functions. An analogous isomorphism is constructed for the symmetric space of symplectic forms over a finite field, with the spherical functions being sent to Macdonald polynomials with parameters $(q,q^2)$. An analogue of parabolic induction is interpreted as a certain multiplication of symmetric functions. Applications are given to Schur positivity of skew Macdonald polynomials with parameters $(q,q^2)$ as well as combinatorial formulas for spherical function values.

Conjectures and results about parabolic induction of representations of $${\text {GL}}_n(F)$$

In 1980 Zelevinsky introduced certain commuting varieties whose irreducible components classify complex, irreducible representations of the general linear group over a non-archimedean local field with a given supercuspidal support. We formulate geometric conditions for certain triples of such components and conjecture that these conditions are related to irreducibility of parabolic induction. The conditions are in the spirit of the Geiss–Leclerc–Schröer condition that occurs in the conjectural characterization of $$\square $$ □ -irreducible representations. We verify some special cases of the new conjecture and check that the geometric and representation-theoretic conditions are compatible in various ways.

Modulo p parabolic induction of pro-p-Iwahori Hecke algebra

We study the structure of parabolic inductions of a pro-p-Iwahori Hecke algebra. In particular, we give a classification of irreducible modulo p representations of pro-p-Iwahori Hecke algebras in terms of supersingular representations. Since supersingular representations are classified by Ollivier and Vignéras, it completes the classification of irreducible modulo p representations.

Decomposition Rules for the Ring of Representations of Non-Archimedean GLn

Let $\mathcal{R}$ be the Grothendieck ring of complex smooth finite-length representations of the sequence of p-adic groups $\{GL_n(F)\}_{n=0}^\infty $, with multiplication defined through parabolic induction. We study the problem of the decomposition of products of irreducible representations in $\mathcal{R}$. We obtain a necessary condition on irreducible factors of a given product by introducing a width invariant. Width $1$ representations form the previously studied class of ladder representations. We later focus on the case of a product of two ladder representations, for which we establish that all irreducible factors appear with multiplicity one. Finally, we propose a general rule for the composition series of a product of two ladder representations and prove its validity for cases in which the irreducible factors correspond to smooth Schubert varieties.

Springer correspondence for the split symmetric pair in type

In this paper we establish Springer correspondence for the symmetric pair $(\text{SL}(N),\text{SO}(N))$ using Fourier transform, parabolic induction functor, and a nearby cycle sheaf construction. As an application of our results we see that the cohomology of Hessenberg varieties can be expressed in terms of irreducible representations of Hecke algebras of symmetric groups at $q=-1$. Conversely, we see that the irreducible representations of Hecke algebras of symmetric groups at $q=-1$ arise in geometry.