A homological interpretation of Jantzen’s sum formula

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 GLn (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 GLn 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.