Canonical basic sets for split Hecke specializations

Let $O$ be a Noetherian commutative local domain, with fraction field $K$, maximal ideal $\mathfrak{m}$ and residue field $k_0=O\mathrm{/}\mathfrak{m}$. Allow a specified field extension $k\mathrm{/}{k}_0$ as the specialization field. The nonfield DVR case is included: a nonfield discrete valuation ring is equivalently a local PID with nonzero maximal ideal (Equivalent characterizations of a DVR); the original residue-field specialization is $k=k_0$. Let $H_O$ be a finite free $O$-algebra, so it has finite rank as an $O$-module. Assume the generic algebra $H_K=K{\otimes }_O{H}_O$ is split semisimple, meaning a finite product of full matrix algebras over $K$; assume $H_k=k{\otimes }_O{H}_O$ is split, meaning each simple module remains simple after extending the field. A simple module is a nonzero module with no proper nonzero submodule. Write $\mathrm{Irr}(H_K)$ and $\mathrm{Irr}(H_k)$ for their isomorphism classes of simple modules. For each $E\in \mathrm{Irr}(H_K)$, choose an $H_O$-stable $O$-lattice ${\mathrm{\Lambda }}_E\subset E$, meaning a finite free $O$-submodule spanning $E$ over $K$. Assume each simple $M\in \mathrm{Irr}(H_k)$ occurs in at least one reduction $k{\otimes }_{k_0}({\mathrm{\Lambda }}_E\mathrm{/}\mathfrak{m}{\mathrm{\Lambda }}_E)$, so every column below is nonzero. For such $M$, let

$$d_{E,M}=[k{\otimes }_{k_0}({\mathrm{\Lambda }}_E\mathrm{/}\mathfrak{m}{\mathrm{\Lambda }}_E):M]$$

be the multiplicity of $M$ among the composition factors of this finite-dimensional $H_k$-module. A composition series is a finite strict submodule chain with simple successive quotients; one exists by induction on module dimension, and Jordan–Hölder makes its multiplicities independent of the chosen series (Composition series and composition factors of an object, Jordan-Holder theorem in an abelian category). The matrix $D=(d_{E,M})$ is the decomposition matrix for these chosen lattices. This definition makes no choice-independence assertion for arbitrary local orders. The finite-Weyl theorem below supplies free integral models and proves their multiplicities independent in its complete local setting; the rank-one worked specialization gives its lattices directly.

Fix an integer-valued statistic ${\alpha }_E$ on $\mathrm{Irr}(H_K)$. In Geck's finite-Weyl Hecke convention this is the Schur-element exponent denoted ${\alpha }_E$; its particular values do not enter the identity-matrix case. Put

$${\check{\alpha }}_M=\min \{{\alpha }_E:d_{E,M}>0\}\mathrm{.}$$

A subset $B\subseteq \mathrm{Irr}(H_K)$ is a canonical basic set (relative to $D$ and $\alpha$) if there is a bijection $M\mapsto E(M)$ from $\mathrm{Irr}(H_k)$ to $B$ satisfying

$$d_{E(M),M}=1,{\alpha }_{E(M)}={\check{\alpha }}_M,$$

and, for every $E,M$,

$$d_{E,M}>0⟹{\check{\alpha }}_M<{\alpha }_E\text{ or }E=E(M)\mathrm{.}$$

This is a definition only; it makes no claim that such a subset exists in every specialization.