ag.algebraic geometry - What is the Hirzebruch-Riemann-Roch formula for the flag variety of a Lie algebra?

Riemann-Roch for the flag variety is the Weyl Character formula!

More specifically, let $L$ be an ample line bundle on $G/B$, corresponding to the weight $lambda$. According to Borel-Weil-Bott, $H^0(G/B,L)$ is $V_{lambda}$, the irreducible representation of $G$ with highest weight $lambda$, and $H^i(G/B,L)=0$ for $i>0$. So the holomorphic Euler characteristic of $L$ is $mathrm{dim} V_{lambda}$.

As we will see, computing the holomorphic Euler characteristic of $L$ by Hirzebruch-Riemann-Roch gives the Weyl character formula for $mathrm{dim} V_{lambda}$.

Notation:

$G$ is a simply-connected semi-simple algebraic group, $B$ a Borel and $T$ the maximal torus in $B$. The corresponding Lie algebras are $mathfrak{g}$, $mathfrak{b}$, $mathfrak{t}$. The Weyl group is $W$, the length function on $W$ is $ell$ and the positive roots are $Phi^{+}$. It will simplify many signs later to take $B$ to be a lower Borel, so the weights of $T$ acting on $mathfrak{b}$ are $Phi^{-}$.

We will need notations for the following objects:
$$rho = (1/2) sum_{alpha in Phi^{+}} alpha.$$
$$Delta = prod_{alpha in Phi^{+}} alpha.$$
$$delta = prod_{alpha in Phi^{+}} (e^{alpha/2}-e^{-alpha/2}).$$

They respectively live in $mathfrak{t}^*$, in the polynomial ring $mathbb{C}[mathfrak{t}^*]$ and in the power series ring $mathbb{C}[[mathfrak{t}^*]]$.

Geometry of flag varieties

Every line bundle $L$ on $G/B$ can be made $G$-equivariant in a unique way. Writing $x$ for the point $B/B$, the Borel $B$ acts on the fiber $L_x$ by some character of $T$. This is a bijection between line bundles on $G/B$ and characters of $T$. Taking chern classes of line bundles gives classes in $H^2(G/B)$. This extends to an isomorphism $mathfrak{t}^* to H^2(G/B, mathbb{C})$ and a surjection $mathbb{C}[[mathfrak{t}^*]] to H^*(G/B, mathbb{C})$. We will often abuse notation by identifiying a power series in $mathbb{C}[[mathfrak{t}^*]]$ with its image in $H^*(G/B)$.

We will need to know the Chern roots of the cotangent bundle to $G/B$.
Again writing $x$ for the point $B/B$, the Borel $B$ acts on the tangent space $T_x(G/B)$ by the adjoint action of $B$ on $mathfrak{g}/mathfrak{b}$. As a $T$-representation, $mathfrak{g}/mathfrak{b}$ breaks into a sum of one dimensional representations, with characters the positive roots. We can order these summands to give a $B$-equivariant filtration of $mathfrak{g}/mathfrak{b}$ whose quotients are the corresponding characters of $B$. Translating this filtration around $G/B$, we get a filtration on the tangent bundle whose associated graded is the direct sum of line bundles indexed by the negative roots. So the Chern roots of the tangent bundle are $Phi^{+}$. (The signs in this paragraph would be reversed if $B$ were an upper Borel.)

The Weyl group $W$ acts on $mathfrak{t}^*$. This extends to an action of $W$ on $H^*(G/B)$. The easiest way to see this is to use the diffeomorphism between $G/B$ and $K/(K cap T)$, where $K$ is a maximal compact subgroup of $G$; the Weyl group normalizes $K$ and $T$ so it gives an action on $K/(K cap T)$.

We need the following formula, valid for any $h in mathbb{C}[[mathfrak{t}^*]]$:
$$int h = mbox{constant term of}left( (sum_{w in W} (-1)^{ell(w)} w^*h)/Delta right). quad (*)$$
Two comments: on the left hand side, we are considering $h in H^*(G/B)$ and using the standard notation that $int$ means "discard all components not in top degree and integrate." On the right hand side, we are working in $mathbb{C}[[mathfrak{t}^*]]$, as $Delta$ is a zero divisor in $H^*(G/B)$.

Sketch of proof of (*): The action of $w$ is orientation reversing or preserving according to the sign of $ell(w)$. So $int h = int (sum_{w in W} (-1)^{ell(w)} w^*h) / |W|$. Since the power series $sum_{w in W} (-1)^{ell(w)} w^*h$ is alternating, it is divisible by $Delta$ and must be of the form $Delta(k + (mbox{higher order terms}))$ for some constant $k$. The higher order terms, multiplied by $Delta$, all vanish in $H^*(G/B)$, so we have $int h = k int Delta/|W|$. The right hand side of $(*)$ is just $k$.

By the Chern root computation above, the top chern class of the tangent bundle is $Delta$. So $int Delta$ is the (topological) Euler characteristic of $G/B$. The Bruhat decomposition of $G/B$ has one even-dimensional cell for every element of $W$, and no odd cells, so $int Delta = |W|$ and we have proved formula $(*)$.

The computation

We now have all the ingredients. Consider an ample line bundle $L$ on $G/B$, corresponding to the weight $lambda$ of $T$. The Chern character is $e^{lambda}$. HRR tells us that the holomorphic Euler characteristic of $L$ is
$$int e^{lambda} prod_{alpha in Phi^{+}} frac{alpha}{1 - e^{- alpha}}.$$
Elementary manipulations show that this is
$$int frac{ e^{lambda + rho} Delta}{delta}.$$

Applying $(*)$, and noticing that $Delta/delta$ is fixed by $W$, this is
$$mbox{Constant term of} left( frac{1}{Delta} frac{Delta}{delta} sum_{w in W} (-1)^{ell(w)} w^* e^{lambda + rho} right)=$$
$$mbox{Constant term of} left( frac{sum_{w in W} (-1)^{ell(w)} e^{w(lambda + rho)}}{delta} right).$$

Let $s_{lambda}$ be the character of the $G$-irrep with highest weight $lambda$. By the Weyl character formula, the term in parentheses is $s_{lambda}$ as an element of $mathbb{C}[[mathfrak{t}^*]]$. More precisely, a character is a function on $G$. Restrict to $T$, and pull back by the exponential to get an analytic function on $mathfrak{t}$. The power series of this function is the expression in parentheses. Taking the constant term means evaluating this character at the origin, so we get $dim V_{lambda}$, as desired.

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