For any Coxeter group, Soergel gave a straightforward construction of a collection of bimodules, now called Soergel bimodules, over the coordinate ring of the reflection representation. Soergel bimodules form a monoidal category, whose Grothendieck ring is isomorphic to the Hecke algebra of the Coxeter group. Soergel conjectured that (when defined over a field of characteristic zero) the indecomposable bimodules would descend to the Kazhdan-Lusztig basis, which would give an algebraic proof of the various positivity conjectures put forth by Kazhdan and Lusztig.

For Weyl groups, Soergel bimodules are constructed to agree precisely with the equivariant intersection cohomology of Schubert varieties. That is, Soergel bimodules are (by definition) summands of Bott-Samelson bimodules, which are the equivariant cohomology of Bott-Samelson resolutions of Schubert varieties. Therefore, the Decomposition theorem implies that the indecomposable bimodules agree with the intersection cohomology of the simple perverse sheaves on the flag variety. However, without the use of the Decomposition theorem, there is no a priori reason why the Bott-Samelson bimodules should split into summands as expected.

Inspired by de Cataldo and Migliorini's Hodge-theoretic proof of the Decomposition Theorem, we provide an algebraic proof of the Soergel conjecture for a general Coxeter group. Moreover, we show algebraically that Soergel bimodules have the Hodge-theoretic properties expected of an equivariant intersection cohomology space. This is joint work with Geordie Williamson.

If time permits, we will advertise a counter-example to the original version of Soergel's conjecture: a quantized version of the geometric Satake equivalence.