A minimal even type of the 2-adic Weil representation

TL;DR

This paper constructs a minimal type of the 2-adic Weil representation for symplectic groups over _2 and shows its Hecke algebra is isomorphic to a classical affine Hecke algebra of an orthogonal group.

Contribution

It introduces a minimal type for the 2-adic Weil representation and establishes an isomorphism with a classical affine Hecke algebra, linking representation theory and algebraic structures.

Findings

Hecke algebra is isomorphic to the affine Hecke algebra of _2's split orthogonal group.

Constructs a minimal type for the 2-adic Weil representation.

Provides new connections between symplectic and orthogonal group representations.

Abstract

The Weil representation is used to construct a minimal type of the two-fold central extension of $\operatorname{Sp}_{2n}(\mathbb{Q}_2)$. The corresponding Hecke algebra is shown to be isomorphic to the classical affine Hecke algebra of the split adjoint orthogonal group $\operatorname{SO}_{2n+1}(\mathbb{Q}_2)$.