Kazhdan–Lusztig conjecture

Remarks

Kazhdan and Lusztig stated the conjecture in 1979. It was proved in 1981, independently by Beilinson and Bernstein and by Brylinski and Kashiwara. Both proofs pass to $\mathcal{D}$-modules on the flag variety and use the Riemann–Hilbert correspondence. There, $P_{y,w}$ is a generating function for the local intersection cohomology of Schubert varieties, a result of Kazhdan and Lusztig. An equivalent form gives the multiplicities in category O: $[M_w : L_y] = P_{w_0 w, w_0 y}(1)$, where $w_0$ is the longest element of $W$. Example: for $\mathfrak{sl}_2(\C)$, $W = \set{1, s}$ and $P_{1,s} = P_{1,1} = P_{s,s} = 1$. Here $M_1 = M(-2)$ is irreducible and $M_s = M(0)$ contains $M(-2)$, so $\operatorname{ch} L_s = \operatorname{ch} M_s - \operatorname{ch} M_1$, as the formula says.

Going deeper

For a student

For $w = w_0$ the highest weight $-w_0\rho - \rho = 0$ is dominant, all $P_{y,w_0}$ equal $1$, and the formula gives the trivial character as in the Weyl character formula. The other $L_w$ are infinite-dimensional, and for them no earlier formula was known in general. Through translation functors, it gives the characters of all simple modules of integral weight in category $\mathcal{O}$. Its proof was a founding success of geometric representation theory.