arXiv subjects

Elmar Grosse-Klönne

Publications and source records attributed to Elmar Grosse-Klönne.

21 records · Page 2Linked to original sources

arXiv2014

On the universal module of $p$-adic spherical Hecke algebras

Let $\widetilde{G}$ be a split connected reductive group with connected center $Z$ over a local non-Archimedean field $F$ of residue characteristic $p$, let $\widetilde{K}$ be a hyperspecial maximal compact open subgroup in $\widetilde{G}$. Let $R$ be a commutative ring, let $V$ be a finitely generated $R$-free $R[\widetilde{K}]$-module. For an $R$-algebra $B$ and a character $χ:{\mathfrak H}_V(\widetilde{G},\widetilde{K})\to B$ of the spherical Hecke algebra ${\mathfrak H}_V(\widetilde{G},\widetilde{K})={\rm End}_{R[\widetilde{G}]}{\rm ind}_{\widetilde{K}}^{\widetilde{G}}(V)$ we consider the specialization $$M_χ(V)={\rm ind}_{\widetilde{K}}^{\widetilde{G}}V\otimes_{{\mathfrak H}_V(\widetilde{G},\widetilde{K}),χ}B$$ of the universal ${\mathfrak H}_V(\widetilde{G},\widetilde{K})$-module ${\rm ind}_{\widetilde{K}}^{\widetilde{G}}(V)$. For large classes of $R$ (including ${\mathcal O}_F$ and $\overline{\mathbb F}_p$), $V$, $B$ and $χ$, arguing geometrically on the Bruhat Tits building we give a sufficient criterion for $M_χ(V)$ to be $B$-free and to admit a $\widetilde{G}$-equivariant resolution by a Koszul complex built from finitely many copies of ${\rm ind}_{\widetilde{K}Z}^{\widetilde{G}}(V)$. This criterion is the exactness of certain fairly small and explicit ${\mathfrak N}$-equivariant $R$-module complexes, where ${\mathfrak N}$ is the group of ${\mathcal O}_F$-valued points of the unipotent radical of a Borel subgroup in $\widetilde{G}$. We verify it if $F={\mathbb Q}_p$ and if $V$ is an irreducible $\overline{\mathbb F}_p[\widetilde{K}]$-representation with highest weight in the (closed) bottom $p$-alcove, or a lift of it to ${\mathcal O}_F$. We use this to construct $p$-adic integral structures in certain locally algebraic representations of $\widetilde{G}$.

math.RT↗

arXiv2014

Locally unitary principal series representations of ${\rm GL}_{d+1}(F)$

For a local field $F$ we consider tamely ramified principal series representations $V$ of $G={\rm GL}_{d+1}(F)$ with coefficients in a finite extension $K$ of ${\mathbb Q}_p$. Let $I_0$ be a pro-$p$-Iwahori subgroup in $G$, let ${\mathcal H}(G,I_0)$ denote the corresponding pro-$p$-Iwahori Hecke algebra. If $V$ is locally unitary, i.e. if the ${\mathcal H}(G,I_0)$-module $V^{I_0}$ admits an integral structure, then such an integral structure can be chosen in a particularly well organized manner, in particular its modular reduction can be made completely explicit.

math.RT↗

arXiv2014

On special representations of $p$-adic reductive groups

Let $F$ be a non-Archimedean locally compact field, let $G$ be a split connected reductive group over $F$. For a parabolic subgroup $Q\subset G$ and a ring $L$ we consider the $G$-representation on the $L$-module$$(*)\quad\quad\quad\quad C^{\infty}(G/Q,L)/\sum_{Q'\supsetneq Q}C^{\infty}(G/Q',L).$$Let $I\subset G$ denote an Iwahori subgroup. We define a certain free finite rank $L$-module ${\mathfrak M}$ (depending on $Q$; if $Q$ is a Borel subgroup then $(*)$ is the Steinberg representation and ${\mathfrak M}$ is of rank one) and construct an $I$-equivariant embedding of $(*)$ into $C^{\infty}(I,{\mathfrak M})$. This allows the computation of the $I$-invariants in $(*)$. We then prove that if $L$ is a field with characteristic equal to the residue characteristic of $F$ and if $G$ is a classical group, then the $G$-representation $(*)$ is irreducible. This is the analog of a theorem of Casselman (which says the same for $L={\mathbb C}$); it had been conjectured by Vignéras. Herzig (for $G={\rm GL}_n(F)$) and Abe (for general $G$) have given classification theorems for irreducible admissible modulo $p$ representations of $G$ in terms of supersingular representations. Some of their arguments rely on the present work.

math.RT↗