indecomposable modules restricted from $gl_n$ to $sl_n$

Question

Let K be an algebraic closed field, $gl_n$ be the general linear Lie algebra over K, and $sl_n$ be the special linear Lie algebra. Let $\chi\in gl_n^*$. Let $U_\chi(gl_n)$ be the corresponding reduced enveloping algebra for $gl_n$. If M is an indecomposable module for $U_\chi(gl_n)$, then is M still indecomposable as a module for $U_\chi(sl_n)$? I think it is still true. But I think it is not easy to prove it. Need Help.

Actually, I am considering that the similar question comes out naturally for a more general setting. Let A be a finite dimensional algebra over k, B is a sub algebra. Then can we find a condition for B such that all indecomposable (finite dimensional ) module for A is still indecomposable (finite dimensional ) module when restricted to B?

Answer 1

This is actually true (in somewhat more generality), as remarked by Jantzen in a recent updating of his unpublished 2011 notes on restrictions of modular representations of $\mathfrak{gl}_n$ to $\mathfrak{sl}_n$. Here he works with representations attached to reduced enveloping algebras relative to various linear functionals $\chi$ on the two Lie algebras. (But his results are formulated more generally for certain pairs of Lie algebras related in a way similar to these two.)

He points out in his 2.6 that Theorem 2.1.2 in an earlier paper by Farnsteiner here generalizes easily to show that the restriction of an indecomposable module remains indecomposable for any $\chi$ (not just for $\chi=0$ as in Farnsteinber's work on group schemes).

Jantzen's main focus is actually on simple modules and their projective covers, where for $p=n=3$ there is some anomalous behavior of the restrictions (one of the cases necessarily excluded from Premet's proof of the Kac-Weisfeiler conjecture on powers of $p$ dividing dimensions of simple modules). This particular case was worked out concretely in a thesis by Xin WEN in Shanghai (write-up to appear in J. Pure Appl. Algebra), which partly motivated Jantzen's more general study. Along the way it becomes clear that the indecomposable projectives remain indecomposable on restriction, which is seen more directly than in Farnsteiner's set-up.