# Friday, October 08

:::{.remark}
Continuing some stuff from Kumar Ch. 8: the goal is to understand the Demazure and Weyl-Kac character formulas.
Open question: how can one compute the singular locus of a given Schubert variety?
This is surprisingly a hot topic this semester, c/o multiple Arxiv papers that have come out over the past few months.

Our first goal: showing $X_w^Y$ is normal.
Note that most varieties in representation theory are not normal, and this complicates things significantly, so normality is a great condition here.

Recall that for $X\in \Var$, the stalks $\OO_{X, x}$ are local rings, and the **cotangent space at $x$** is defined as $\mfm_x/\mfm_x^2$.

> Cohomology vanishing: some of the hardest and most important results in this area!

:::

:::{.theorem title="8.1.8, Main Result"}
Let $w = (s_{i_1} \cdots, s_{i_n}) \in W$ be a word and consider $j, k$ such that $1\leq j\leq k\leq n$.
Suppose that the subword $v = (s_{i_j} \cdots, s_{i_k})$ is reduced.
Considering the associated BSDH-varieties, we have a subvariety
\[
Z_v \da P_{i_j}\mix{B} \cdots P_{i_k}/B \injects
Z_w \da P_{i_1}\mix{B} \cdots P_{i_n}/B
.\]
Recall that $\mcl^Y(\lambda) \da G\mix{P_Y} \CC_{- \lambda}$, and 
\[
\mcl_w( \lambda) \da P_{i_1}\mix{B} \cdots \mix{B} P_{i_n} \CC_{- \lambda} = f^* \mcl^Y( \lambda)
,\] 
and we write $w(n)$ for $w$ with the $n$th letter omitted.
Moreover codimension 1 subvarieties correspond to line bundles under the Chern class isomorphism.
Then for any integral dominant $\lambda \in D_\ZZ$, there are 3 vanishing formulas:

1. 
\[
H^{\geq 1 } \qty{ Z_w; \mcl_w( \lambda) \tensor \OO_{Z_w}(- \sum_{q=0}^k Z_{w(q)} ) } &= 0 
.\]

2.
\[
H^{\geq 1} \qty{Z_w; \mcl_w( \lambda) } &= 0 
.\]

3. If $k<n$ and $v' \da (s_{i_k} \cdots, s_{i_k} s_{i_{k+1}})$ is *not* reduced, then

\[
H^{\geq 0 } \qty{ Z_w; \mcl_w( \lambda) \tensor \OO_{Z_v}(- \sum_{q=j}^k Z_{w(q)} ) } &= 0 \\
.\]


:::

:::{.remark}
We'll often use **Serre duality** in the following form:
given a set of nice assumptions, there is a perfect pairing
\[
H^i(X; \mcf) \times H^{n-i}(X; K_X \tensor \mcf\dual) \to \CC
,\]
where $\mcf\dual \da \Hom_{\OO_X}(\mcf, \OO_X)$ and $K_X$ is the dualizing sheaf.
Note that if $X$ is smooth and projective, one can take $K_X$ to be the canonical sheaf.
:::

:::{.lemma title="8.18"}
For any finite-dimensional representation $M$ of $B\cartpower{n}$, there is a functorial assignment a $P_{i_1}\dash$equivariant algebraic vector bundle $\mcl_w(M) \to Z_w$ on the BSDH variety.
which is an exact functor on $\mods{B\cartpower{n}}^{\dim < \infty}$, given by
\[
\mcl_w( - \lambda ) = \mcl_w( \CC_{ \lambda})
.\]
This is induced by $B\cartpower{n} \mapsvia{\pr_n} B \to \CC_{\lambda}$.
:::

:::{.remark}
Using this formula, 
\[
\mcl(\lambda)\dual = \mcl( \CC_{- \lambda})\dual = \mcl(\CC_{- \lambda} \dual)
,\]
where given $V_{ \lambda}$ a highest weight representation of $G$ (connected reductive finite type), we have $V_{ \lambda}\dual = V_{-w_0 \lambda}$.
Here $\CC_{ - \lambda}$ is a representation of the torus, for which $w_0 = \id$.
:::

:::{.example title="?"}
For $w = v = (s)$ and $1\leq j\leq k \leq n =1$, we have $Z_{(s)} = P_s/B \cong \PP^1$.
Noting $Z_{ \emptyset} = B/B = \pt$, using the formula we obtain
\[
H^p( P_s/B; \mcl_{(s)} ( \lambda) \tensor \OO_{Z_{(s)}} (-B/B) )
.\]
This is an (equivariant) line bundle on $\PP^1$, which are all of the form $\OO(n)$ -- which one is it?
Write $a \da \ip{ \lambda}{ \alpha\dual} \in \ZZ_{\geq 0}$ since $\lambda$ was dominant integral.
Forgetting the group action yields an algebraic bundle which we can write as
\[
H^p(P_s/B ; \OO_{\PP^1}(a) \tensor \OO_{\PP^1}(-1) )
.\]

This can also be described by tensoring $\CC_{ \lambda} \tensor \CC_{ \mu} \cong \CC_{ \lambda+ \mu}$.
Finally, we can identify this homology with
\[
H^p(P_s/B ; \OO_{\PP^1}(a-1) )
.\]


Note that the canonical for $\PP^1$ is $G\mix{B} \CC_{2} = \mcl(-2) = \OO(-2)$ (noting that $-n$ has no global sections).
So if $\mcf = \OO(k)$ then $\mcf\dual = \OO(-k)$.
Applying Serre duality yields
\[
H^i( \PP^1; \OO(k) ) \times H^{n-i}(\PP^1; \OO(-2) \tensor \OO(-k) ) \to \CC
.\]

Note that $H^0(\PP^1; \OO(k)) = 0$ for $k<0$, since these look like homogeneous polynomials in degree $k$ (and there are none of negative degree), so taking $k=-1$ we have $H^0(\PP^1; \OO(-1)) = 0$.
By duality, this pairs with $H^1(\PP^1; \OO(-1))$, and continuing yields a pairing:

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:::

:::{.example title="?"}
Let $w=(s, s), v=(s), v'=(s), \eta = \emptyset, \mu = \emptyset$.
Write $\sigma = \psi: Z_w \to Z_v$, which is projection onto the first coordinates in the corresponding BSDH varieties:
\[
P_s \mix{B} P_s/B &\to P_s/B \\
[p_1, p_2B/B] &\mapsto [p_1 B/B]
.\]
Note that 
\[
\OO_{Z_w}( -B\mix{B} P_s /B) = \sigma^* ( \OO_{Z_v}(-B/B) )
.\]
There are 3 important facts we'll revisit:

1. A projection formula,
2. Lemma 8.1.5,
3. The Leray spectral sequence.

:::