Propri\'et\'es de maximalit\'e concernant une repr\'esentation d\'efinie par Lusztig

TL;DR

This paper investigates the properties of maximal elements in the decomposition of certain Weyl group representations associated with symplectic partitions, extending known minimality results to maximality under specific conditions.

Contribution

It proves the existence of a maximal pair in the decomposition of the generalized Springer correspondence for symplectic partitions with only even parts.

Findings

Existence of a maximal pair $(mbda^{max},psilon^{max})$ with positive multiplicity.

The maximal pair appears with multiplicity one in the decomposition.

The maximal pair dominates all other pairs in the partial order.

Abstract

Let $\lambda$ be a symplectic partition, denote Jord^{bp}($\lambda$) the set of even positive integers i which appear in $\lambda$, and let a map $\epsilon:Jord^{bp}(\lambda) \to {\pm 1}$. The generalized Springer's correspondence associates to $(\lambda,\epsilon)$ an irreducible representation $\rho(\lambda,\epsilon)$ of some Weyl group. We can also define a representation $\underline{\rho}(\lambda,\epsilon)$ of the same Weyl group, in general reducible. Roughly speaking, $\rho(\lambda,\epsilon)$ is the representation of the Weyl group in the top cohomology group of some variety and $\underline{\rho}$ is the representation in the sum of all the cohomology groups of the same variety. The representation $\underline{\rho}$ decomposes as a direct sum of $\rho(\lambda',\epsilon')$ with some multiplicities, where $(\lambda',\epsilon')$ describes the pairs similar to $(\lambda,\epsilon)$. It is well know that $(\lambda,\epsilon)$ appears in this decomposition with multiplicity one and is minimal in this decomposition. That is, if $(\lambda',\epsilon')$ appears, we have $\lambda'>\lambda$ or $(\lambda',\epsilon')=(\lambda,\epsilon)$. Assuming that $\lambda$ has only even parts, we prove that there exists also a maximal pair $(\lambda^{max},\epsilon^{max})$. That is $4(\lambda^{max},\epsilon^{max})$ appears with positive multiplicity (in fact one) and, if $(\lambda',\epsilon')$ appears, we have $\lambda^{max}>\lambda'$ or $(\lambda',\epsilon')=(\lambda^{max},\epsilon^{max})$.