A homological interpretation of Jantzen's sum formula

Abstract

For a split reductive algebraic group, this paper observes a homological interpretation for Weyl module multiplicities in Jantzen's sum formula. This interpretation involves an Euler characteristic χ built from Ext groups between integral Weyl modules. The new interpretation makes transparent For GL n (and conceivable for other classical groups) a certain invariance of Jantzen's sum formula under "Howe duality" in the sense of Adamovich and Rybnikov. For GL n a simple and explicit general formula is derived for χ between an arbitrary pair of integral Weyl modules. In light of Brenti's work on certain R-polynomials, this formula raises interesting questions about the possibility of relating Ext groups between Weyl modules to Kazhdan-Lusztig combinatorics. IntroductionLet G Z be a split and connected reductive algebraic group scheme over Z. For a prime number p, G p will denote the corresponding group scheme over F p , the field of p elements. We will be concerned with (rational) representations of G p , in the course of which we will need to use G Z . We will need some background material on these topics. For all such standard facts see the recent edition of Jantzen's classic text [Jantzen], where one can also find the original references.For a finite dimensional rational representation M of G p , its formal character isW , the Weyl group invariants in the integral group ring of the character group X of a split maximal torus. A central problem is to calculate formal characters of all simple modules, which are in bijective correspondence with their highest weights. To get to the issue of interest in this paper, first let L p (λ) and V p (λ) respectively be the simple module and the Weyl module (i.e., the universal highest weight module) corresponding to a given dominant integral weight λ. Since each of the families {ch(L p (λ)} and {ch(V p (λ))} forms a basis of Z [X] W , and since ch(V p (λ)) is known by Weyl's character formula, the problem 1

Sum formulas for reductive algebraic groups

“…The second identity in this corollary was obtained by the second author in [14]. The argument there is different.…”

“…At the meeting AMS Scand 2000 in Odense, Denmark the second author gave a talk, "Ext groups and Jantzen's sum formula" in which he presented the Weyl module sum formula in terms of Ext-groups. This can be found in [14], and it is also referred to in the preprint [15] where he proves the equivalence with the sum formula for tilting modules. Shortly after the appearance of this preprint we realized how to give the uniform proof presented below.…”

“…In this paper we start out by proving an Euler-type formula for such Ext-groups using techniques from [1] and [14]. Then we are able to deduce the two sum formulas from this.…”

“…Both the above mentioned sum formulas are related to Ext-groups involving integral versions of Weyl modules, see [14] and [3]. In this paper we start out by proving an Euler-type formula for such Ext-groups using techniques from [1] and [14].…”

“…We have taken the opportunity to recall the arguments from [1,3,14] that we need. In this way our proof of the sum formulas for G is completely self-contained relying only on basic facts on Weyl modules, cohomology on line bundles, and tilting modules (which can all be found in [12]).…”