Abstract
HarishChandra induction and restriction functors play a key role in the representation theory of reductive groups over finite fields. In this paper, extending earlier work of Dat, we introduce and study generalisations of these functors which apply to a wide range of finite and profinite groups, typical examples being compact open subgroups of reductive groups over nonarchimedean local fields. We prove that these generalisations are compatible with two of the tools commonly used to study the (smooth, complex) representations of such groups, namely Clifford theory and the orbit method. As a test case, we examine in detail the induction and restriction of representations from and to the Siegel Levi subgroup of the symplectic group $\text{Sp}_{4}$ over a finite local principal ideal ring of length two. We obtain in this case a Mackeytype formula for the composition of these induction and restriction functors which is a perfect analogue of the wellknown formula for the composition of HarishChandra functors. In a different direction, we study representations of the Iwahori subgroup $I_{n}$ of $\text{GL}_{n}(F)$ , where $F$ is a nonarchimedean local field. We establish a bijection between the set of irreducible representations of $I_{n}$ and tuples of primitive irreducible representations of smaller Iwahori subgroups, where primitivity is defined by the vanishing of suitable restriction functors.
Original language  English 

Pages (fromto)  9931049 
Number of pages  57 
Journal  Journal of the Institute of Mathematics of Jussieu 
Volume  18 
Issue number  5 
Early online date  14 Aug 2017 
DOIs  
Publication status  Published  Sep 2019 
Keywords
 HarishChandra induction
 parabolic induction
 compact padic groups
 representations of profinite groups
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Ehud Meir Ben Efraim
 School of Natural & Computing Sciences, Mathematical Science  Senior Lecturer
Person: Academic