Finite local systems in the Drinfeld-Laumon construction

TL;DR

This paper analyzes the behavior of finite local systems under the Drinfeld-Laumon construction, revealing that under certain conditions, the resulting sheaf corresponds to the IC sheaf of a specific Harder-Narasimhan stratum.

Contribution

It establishes explicit conditions on local systems' monodromy for the Drinfeld-Laumon construction to produce IC sheaves of Harder-Narasimhan strata with specified subquotients.

Findings

The construction yields IC sheaves of Harder-Narasimhan strata under given monodromy conditions.

Explicit relations between Young diagram rows and resulting sheaves.

Conditions on monodromy ensure the stability of the sheaf structure.

Abstract

Let E be a local system on a smooth projective curve of genus g with monodromy given by a representation of the symmetric group corresponding to a Young diagram with rows of lengths n_1,n_2,... where n_1 > n_2 + (2g-2), n_2 > n_3 + (2g-2), ..., n_{k-1} > n_k + (2g-2), n_k > n_{k+1} + n_{k+2} + ... + (2g-2). We show that the result of k steps of the Drinfeld-Laumon construction applied to E is the IC sheaf of the Harder-Narasimhan stratum with subquotients of rank 1 and degrees n_1, n_2, ..., n_k, n_{k+1}+n_{k+2}+... with coefficients in a local system with monodromy given by the Young diagram with rows n_{k+1}, n_{k+2}, ...