The classical parabolic induction functor is a fundamental tool on the representation theoretic side of Langlands program. In this article, we study its derived version. It was shown by second author that category smooth G $G$ -representations over k $k$ , p $p$ -adic reductive group and field characteristic equivalent to certain differential graded -algebra H • $H_G^\bullet$ whose zeroth cohom...

David Vogan gave programmatic conjectures about the Dixmier's map and he made two conjectures that induction may be independent of the choice of parabolic group used and the sheets of orbit data are conjugated or disjointed. In our previous paper, we gave a geometric version of the parabolic induction of the geometric orbit datum (i.e. orbit covers), and proved Vogan's rst conjecture for geomet...

By restricting to a special class of smooth functions, the local action of the symmetry group is globalized. This special class of functions is constructed using parabolic induction.

Preface 1 1. Classical groups 4 2. Parabolic induction 10 3. Admissible representations 16 4. Jacquet modules and cuspidal representations 24 5. Composition series of induced representations of SL(2, F ) and GL(2, F ) 34 6. Some examples 39 7. Parabolically induced representations of SL(2, F ) and GL(2, F ) 45 8. Some general consequences 52 9. GL(n, F ) 55 10. GSp(n, F ) 62 11. On the reducibi...

This article gives conceptual statements and proofs relating parabolic induction and Jacquet functors on split reductive groups over a nonArchimedean local field to the associated Iwahori-Hecke algebra as tensoring from and restricting to parabolic subalgebras. The main tool is Bernstein’s presentation of the Iwahori-Hecke algebra.

Let $\mathbf{G}$ be a connected reductive group defined over locally compact non-archimedean field $F$, let $\mathbf{P}$ parabolic subgroup with Levi $\mathbf{M}$ and compatible pro-$p$ Iwahori of $G := \mathbf{G}(F)$. $R$ commutative unital ring. We introduce the Iwahori--Hecke $R$-algebra $\mathcal{H}_R(P)$ $P \mathbf{P}(F)$ construct two morphisms $\Theta^P_M\colon \mathcal{H}_R(P)\to \mathc...