# Strange formulas that gave rise to Koszul duality

According to p.8 of the note KOSZUL DUALITY AND APPLICATIONS IN REPRESENTATION THEORY by Geordie Williamson.

Let $$M(\eta)$$ be the Verma module of weight $$\eta$$, $$L(\eta)$$ be its unique simple quotient and $$w_0$$ be the longest element in $$W$$.

The strange formula in our notation is $$[M(x\cdot 0):L(y\cdot 0)]=\sum_{i\ge 0}\dim\mathrm{Ext}_{\mathcal{O}}^i(M(w_0x\cdot 0):L(w_0y\cdot 0))$$ for $$x,y\in W$$.

Let $$\mu$$ be an integral, antidominant weight, $$\Delta$$ be the set of simple roots, $$\Sigma=\{\alpha\in\Delta:\langle\mu+\rho,\alpha^\lor\rangle=0\}$$ and $$W^{\Sigma}=\{w\in W:w.

Do we still have $$[M(x\cdot \mu):L(y\cdot \mu)]=\sum_{i\ge 0}\dim\mathrm{Ext}_{\mathcal{O}}^i(M(w_0x\cdot \mu):L(w_0y\cdot \mu))$$ for $$x,y\in W^\Sigma$$?

No. If the left-hand side is in the singular block, the right hand-side should be in the regular parabolic block, i.e. its Koszul dual.

(The regular block for the Borel subalgebra is Koszul-self-dual.)

See Beilinson-Ginzburg-Soergel: Koszul Duality Patterns in Representation Theory, J. Amer. Math. Soc. 9 (1996), 473-527.

This is a more concrete explanation for what Rafael Mrđen had said.

Let $$x,w\in W^\Sigma$$ and $$\ell(x,w)=\ell(w)-\ell(x)$$.

It is well-known that $$\mathrm{ch}L(w\cdot\mu)=\sum_{x\in W^\Sigma}(-1)^{\ell(x,w)}P^{\Sigma}_{x,w}(1)\mathrm{ch}M(x\cdot\mu)$$, where $$P^\Sigma_{x,w}(q)=\sum_{i\ge 0}q^{\frac{\ell(x,w)-i}{2}}\dim\mathrm{Ext}_{\mathcal{O}}^i(M(x\cdot\mu),L(w\cdot\mu))$$.

Let $$P_{u,v}(q)$$ be the Kazhdan Lusztig polynomial of $$W$$.

Let $$P_{u,v}^{\Sigma,q}(q)$$ be the parabolic Kazhdan Lusztig polynomial of $$W^\Sigma$$ of type $$q$$.

Let $$P_{u,v}^{\Sigma,-1}(q)$$ be the parabolic Kazhdan Lusztig polynomial of $$W^\Sigma$$ of type $$-1$$.

By Section 8.2 of KOSTANT MODULES IN BLOCKS OF CATEGORY $$\mathcal{O}_S$$ (taking $$S=\emptyset$$, $$J=\Sigma$$), it holds that $$P^\Sigma_{x,w}(q)=\sum_{t\in W_\Sigma}(-1)^{\ell(t)}P_{xt,w}(q)$$.

Note that $$W_\Sigma=\{w\in W: w\cdot\mu=\mu\}=\langle s_\alpha\in W: \alpha\in\Sigma\rangle$$. It is well-known that $$P_{x,w}^{\Sigma,q}(q)=\sum_{t\in W_\Sigma}(-1)^{\ell(t)}P_{xt,w}(q)$$. Hence, $$P^\Sigma_{x,w}(q)=P_{x,w}^{\Sigma,q}(q)$$. Let $$w_\Sigma$$ be the longest element in $$W_\Sigma$$. It is well-known that $$P_{x,w}^{\Sigma,-1}(q)=P_{xw_\Sigma,ww_\Sigma}(q)$$.

Now we try to compute $$[M(x\cdot\mu):L(y\cdot\mu)]$$ by using inverse parabolic Kazhdan Lusztig polynomials.

By Corollary 3.7 (iii) of Hiroyuki Tagawa---Some Properties of Inverse Weighted Parabolic Kazhdan Lusztig Polynomials. It holds that $$\sum_{x\le w\le y, w\in W^\Sigma}(-1)^{\ell(x)+\ell(w)}P_{x,w}^{\Sigma,q}(q)P_{w_0yw_\Sigma,w_0ww_\Sigma}^{\Sigma,-1}(q)=\delta_{x,y}=\begin{cases} 1, x=y\\ 0, x\neq y\\ \end{cases}$$.

Since $$x\not\le w\implies P_{x,w}^{\Sigma,q}(q)=0$$, $$w\not\le y\iff w_0yw_\Sigma\not\le w_0ww_\Sigma\implies P_{w_0yw_\Sigma,w_0ww_\Sigma}^{\Sigma,-1}(q)=0$$ and $$(-1)^{\ell(x)+\ell(w)}=(-1)^{\ell(x,w)}$$. It holds that $$\sum_{w\in W^\Sigma}(-1)^{\ell(x,w)}P_{x,w}^{\Sigma}(q)P_{w_0y,w_0w}(q)=\delta_{x,y}$$.

Then $$\sum_{w\in W^\Sigma}P_{w_0y,w_0w}(1)\mathrm{ch}L(w\cdot\mu)$$ $$=\sum_{w\in W^\Sigma}\sum_{x\in W^\Sigma}(-1)^{\ell(x,w)}P^{\Sigma}_{x,w}(1)P_{w_0y,w_0w}(1)\mathrm{ch}M(x\cdot\mu)$$ $$=\sum_{x\in W^\Sigma}\sum_{w\in W^\Sigma}(-1)^{\ell(x,w)}P^{\Sigma}_{x,w}(1)P_{w_0y,w_0w}(1)\mathrm{ch}M(x\cdot\mu)$$ $$=\sum_{x\in W^\Sigma}\delta_{x,y}\mathrm{ch}M(x\cdot\mu)$$ $$=\mathrm{ch}M(y\cdot\mu).$$

Hence $$[M(y\cdot\mu):L(w\cdot\mu)]=P_{w_0y,w_0w}(1)$$ and then $$[M(x\cdot\mu):L(y\cdot\mu)]$$ $$=P_{w_0x,w_0y}(1)$$ $$=\sum_{i\ge 0}\dim\mathrm{Ext}_\mathcal{O}^i(M(w_0x\cdot(-2\rho)),L(w_0y\cdot(-2\rho))).$$.