Periodicity in filtrations of mod p representations of GL_2(F_q)

Abstract: The irreducible mod $p$ representations of
${GL}_2(\mathbb{F}_p)$ are exactly the twists of $V_r,$ the $r$-th
symmetric power of the standard representations of ${GL}_2(\mathbb{F}_p)$
for small values of $r.$ In this talk, for sufficiently large $r,$ we
investigate the periodicity in a filtration of $V_r$ defined by the powers
of the polynomial $\theta:=X^pY-XY^p,$ motivated by a classical result of
Glover. Ghate and Vangala studied the periodicity of the higher quotients
in the filtration of $V_r$ using generalized dual numbers. We strengthen
their result by defining an explicit isomorphism between these quotients
of $V_r$ and generalized mod $p$ principal series representations using
differential operators, and extend it to ${GL}_2(\mathbb{F}_q)$ for
$q=p^f, f\geq 1.$ In search of a similar periodicity result in case of
cuspidal representations, Reduzzi proved that the reduction mod $p$ of a
cuspidal representation of ${GL}_2(\mathbb{F}_q)$ is isomorphic to the
cokernel of a differential operator on $V_r$ defined by Serre. This
isomorphism is proved using crystalline cohomology and is not explicit. We
define this isomorphism explicitly after tensoring with $V_{q-1.}$ This
work is joint with Eknath Ghate.