# Friday February 7th

So far, we have $\chi_\lambda = \chi_{w.\cdot \lambda}$ if $\lambda \in \Lambda$ and $w\in W$.
We have $\lieh\dual \supset \Lambda$ which is topologically equivalent to $\AA^\ell \supset \ZZ^\ell$, where $\ZZ^\ell$ is dense in the Zariski topology.

For $z\in \mcz(\lieg)$, we have $\chi_\lambda(z) = \chi_{W\cdot \lambda} (z)$ and so $\lambda(\xi(z)) = (w\cdot \lambda )(\xi(z))$ 
where $\xi: \mcz(\lieg) \to U(\lieh) = S(\lieh) \cong P(\lieh\dual)$ where we send $\lambda(f)$ to $f(\lambda)$.

Then $\xi(z)(\lambda) = \xi(z)(w\cdot \lambda)$ for all $\lambda \in \Lambda$, and so $\xi(z) = w\inv \xi(z)$ on $\Lambda$.
But both sides here are polynomials and thus continuous, and $\Lambda \subset \lieh\dual$ is dense, so $\xi(z) = w\inv \xi(z)$ on all of $\lieh\dual$.
I.e., $\chi_\lambda = \chi_{w\cdot } \lambda$ for all $\lambda \in \lieh\dual$.

This in fact shows that the image of $\mcz(\lieg)$ under $\xi$ consists of $W\dash$invariant polynomials.

It's customary to state this in terms of the natural action of $W$ on polynomials without the row-shift.
We do this by letting $\tau_\rho: S(\lieh) \mapsvia{\cong} S(\lieh)$ be the algebra automorphism induced by $f(\lambda) \mapsto f(\lambda - \rho)$.
This is clearly invertible by $f(\lambda) \mapsto f(\lambda + \rho)$.
We then define $$\psi: \mcz(\lieg) \mapsvia{\xi} S(\lieh) \mapsvia{\tau_\rho} S(\lieh)$$ as this composition; this is referred to as the **Harish-Chandra** (HC) homomorphism.

Exercise
: 	Show $\chi_\lambda(z) = (\lambda + \rho) (\psi(z))$ and $\chi_{w\cdot \lambda} (w(\lambda+\rho))(\psi(z))$, where $w(\wait)$ is the usual $w\dash$action.

Replacing $\lambda$ by $\lambda + \rho$ and $w$ by $w\inv$, we get 
$$
w\psi(z) = \psi(z)
$$ 
for all $z\in \mcz(\lieg)$ and all $w\in W$ where $(w\psi(z))(\lambda) = \psi(z)(w\inv \lambda)$.

We've proved that

Theorem (Character Linkage and Image of the HC Morphism)
: \hfill
  a. If $\lambda, \mu \in \lieh\dual$ that are linked, then $\chi_\lambda = \chi_\mu$.

	b. The image of the twisted HC homomorphism $\psi: \mcz(\lieg) \to U(\lieh) = S(\lieh)$ lies in the subalgebra $S(\lieh)^W$.

Example
: Let $\lieg = \liesl_2$.
	Recall from finite-dimensional representations there is a canonical element $c\in \mcz(\lieg)$ called the Casimir element.
	For $\OO$, we need information about the full center $\mcz(\lieg)$ (hence introducing infinitesimal characters).

	Expressing $c$ in the PBW basis yields $c = h^2 + 2h + 4yx$, where $h^2 + 2h \in U(\lieh)$ and $4yx \in \lien^- U(\lieg) + U(\lieg) \lien$.

	> Enveloping algebra convention: $x$s, $h$s, $y$s

	Then $\xi(c) = \pr(c) = h^2 + 2h$, and under the identification $\lieh\dual \iff \CC$ where $\lambda \iff \lambda(h)$, we can identify $\rho \iff \rho(h) = 1$.
	The row shift is given by $\psi(c) = (h-1)^2 + 2(h-1) = h^2 - 1$.
	This is $W\dash$invariant, since $s_{\alpha_h} = -h$.
	But $W = \generators{s_\alpha, 1}$, so $s_\alpha$ generates $W$.

	We also have $\chi_\lambda(c) = (\lambda + \rho) (\psi(c)) = (\lambda+1)^2 - 1$.
	Then 
	$$
	\chi_\lambda(c) = \chi_\mu(c) \iff (\lambda+1)^2 - 1 = (\mu + 1)^2 \iff \mu = \lambda \text{ or } \mu = -\lambda - 2
	$$

	But $\lambda = 1 \cdot \lambda$ and $-\lambda - 2 = s_\alpha \cdot \lambda$, so $\mcz(\lieg) = \generators{c} \definedas \CC[c]$ as an algebra.
	So these characters are equal iff $\mu = w\cdot \lambda$ for $w\in W$.

# Section 1.10: Harish-Chandra's Theorem

Goal: prove the converse of the previous theorem.

Theorem (Harish-Chandra)
: Let $\psi: \mcz(\lieg) \to S(\lieh)$ be the twisted HC homomorphism.
	Then

	a. $\psi$ is an *isomorphism* of $\mcz(\lieg)$ onto $S(\lieh)^W$.

	b. For all $\lambda, \mu \in \lieh\dual$, $\chi_\lambda = \chi_\mu$ iff $\mu = w\cdot \lambda$ for some $w\in W$.

	c. Every central character $\chi: \mcz(\lieg) \to \CC$ is a $\chi_\lambda$.


Proof (of (a))
:  Relies heavily on the *Chevalley Restriction Theorem* (which we won't prove here).

	Initially we have a restriction map on polynomial functions $\theta: P(\lieg) \to P(\lieh)$.
	We identified $P(\lieg) = S(\lieg\dual)$, the formal polynomials on $\lieg\dual$.
	However, for $\lieg$ semisimple, we can identify $S(\lieg\dual) \cong S(\lieg)$ via the Killing form.

	By the Chinese Remainder Theorem, $\theta: S(\lieg)^G \to S(\lieh)^W$ is an isomorphism,
	where the subgroup $G \leq \aut(\lieg)$ is the *adjoint group* generated by $\theset{\exp \ad_x \suchthat x \text{ is nilpotent}}$.

	It turns out that $S(\lieg)^G$ is very close to $\mcz(\lieg)$ -- it is the associated graded of a natural filtration on $\mcz(\lieg)$.
	This is enough to show that $\psi$ is a bijection.

Proof (of (b))
:  We'll prove the contrapositive of the converse.

	Suppose $W\cdot \lambda \intersect W\cdot \mu = \emptyset$ and both are in $\lieh\dual$.
	Since these are finite sets, Lagrange interpolation yields a polynomial that is 1 on $W\cdot \lambda$ and 0 on $W\cdot \mu$.
	Let $g = \frac{1}{\abs W} \sum_{w\in W} w\cdot f$.

	> Note: definitely the dot action here, may be a typo in the book.

	Then $g$ is a $W\cdot$ invariant polynomial with the same properties.
	By part (a), we can get rid of the row shift to obtain an isomorphism $\xi: \mcz(\lieg) \to S(\lieh)^(W\cdot)$, the $W\cdot$ invariant polynomials.
	Choose $z\in \mcz(\lieg)$ such that $\xi(z) = g$, then $\chi_\lambda(z) = \lambda(\xi(z)) = \lambda(g) = g(\lambda) = 1$.
	So $\chi_\mu(z) = 0$ similarly, and $\chi_\lambda = \chi_\mu$.

Proof (of (c))
: This follows from some commutative algebra, we won't say much here.
  Look at maximal ideals in $\CC[x, y,\cdots]$ which correspond to evaluating on points in $\CC^\ell$.

$\qed$

Remark
: Chevalley actually proved that $S(\lieh)^W \cong \CC(p_1, \cdots, p_\ell)$ where the $p_i$ are homogeneous polynomials of degrees $d_1 \leq \cdots \leq d_\ell$.
  These numbers satisfy some remarkable properties:
  $\prod d_i = \abs{W}$ and $d_1 = 2$ (these are called the *degrees of $W$*)


# Section 1.11

Theorem (Category O is Artinian)
: Category $\OO$ is *artinian*, i.e. every $M \in \OO$ is Artinian (DCC) and $\dim \hom_\lieg(M, N) < \infty$ for every $M, N$.

Recall that $\OO$ is known to be Noetherian from an earlier theorem.
This will imply that **every $M$ has a composition/Jordan-Holder series**, so we can take composition factors and multiplicities.

Most interesting question: what are the factors/multiplicities of the simple modules and Verma modules?