Notes on BRST VII: The Harish-Chandra Homomorphism

The Casimir element discussed in the last posting of this series is a distinguished quadratic element of the center Z(\mathfrak g)=U(\mathfrak g)^\mathfrak g (note, here \mathfrak g is a complex semi-simple Lie algebra), but there are others, all of which will act as scalars on irreducible representations. The information about an irreducible representation V contained in these scalars can be packaged as the so-called infinitesimal character of V, a homomorphism

\chi_V: Z(\mathfrak g)\rightarrow \mathbf C

defined by zv=\chi_V(z)v for any z\in Z(\mathfrak g), v\in V. Just as was done for the Casimir, this can be computed by studying the action of Z(\mathfrak g) on a highest-weight vector.

Note: this is not the same thing as the usual (or global) character of a representation, which is a conjugation-invariant function on the group G with Lie algebra \mathfrak g, given by taking the trace of a matrix representation. For infinite dimensional representations V, the character is not a function on G, but a distribution \Theta_V. The link between the global and infinitesimal characters is given by

\Theta_V(zf)=\chi_V(z)\Theta_V(f)

i.e. \Theta_V is a conjugation-invariant eigendistribution on G, with eigenvalues for the action of Z(\mathfrak g) given by the infinitesimal character. Knowing the infinitesimal character gives differential equations for the global character.

The Harish-Chandra Homomorphism

The Poincare-Birkhoff-Witt theorem implies that for a simple complex Lie algebra \mathfrak g one can use the decomposition (here the Cartan subalgebra is \mathfrak h=\mathfrak t_{\mathbf C})

\mathfrak g=\mathfrak h \oplus  \mathfrak n^+ \oplus \mathfrak n^-

to decompose U(\mathfrak g) as

U(\mathfrak g) =U(\mathfrak h) \oplus (U(\mathfrak g)\mathfrak n^+ + \mathfrak n^-U(\mathfrak g))

and show that If z\in Z(\mathfrak g), then the projection of z onto the second factor is in U(\mathfrak g)\mathfrak n^+\cap\mathfrak n^-U(\mathfrak g). This will give zero acting on a highest-weight vector. Defining \gamma^\prime: Z(\mathfrak g)\rightarrow Z(\mathfrak h) to be the projection onto the first factor, the infinitesimal character can be computed by seeing how \gamma^\prime(z) acts on a highest-weight vector.

Remarkably, it turns out that one gets something much simpler if one composes \gamma^\prime with a translation operator

t_\rho: U(\mathfrak h)\rightarrow U(\mathfrak h)

corresponding to the mysterious \rho\in \mathfrak h^*, half the sum of the positive roots. To define this, note that since \mathfrak h is commutative, U(\mathfrak h)=S(\mathfrak h)=\mathbf C[\mathfrak h^*], the symmetric algebra on \mathfrak h, which is isomorphic to the polynomial algebra on \mathfrak h^*. Then one can define

t_\rho (\phi(\lambda))=\phi(\lambda -\rho)

where \phi\in \mathbf C[\mathfrak h^*] is a polynomial on \mathfrak h^*, and \lambda\in\mathfrak h^*.

The composition map

\gamma=t_\rho\circ\gamma^\prime: Z(\mathfrak g)\rightarrow U(\mathfrak h)=\mathbf C[\mathfrak h^*]

is a homomorphism, known as the Harish-Chandra homomorphism. One can show that the image is invariant under the action of the Weyl group, and the map is actually an isomorphism

\gamma: Z(\mathfrak g)\rightarrow \mathbf C[\mathfrak h^*]^W

It turns out that the ring \mathbf C[\mathfrak h^*]^W is generated by dim\ \mathfrak h independent homogeneous polynomials. For \mathfrak g=\mathfrak{sl}(n,\mathbf C) these are of degree 2, 3,\cdots,n (where the first is the Casimir).

To see how things work in the case of \mathfrak g=\mathfrak{sl}(2,\mathbf C), where there is one generator, the Casimir \Omega, recall that

\Omega=\frac{1}{8}h^2 + \frac{1}{4}(ef +fe)=\frac{1}{8}h^2 + \frac{1}{4}(h +2fe)

so one has
\gamma^\prime(\Omega)= \frac{1}{4}(h +\frac{1}{2}h^2)

Here t_\rho(h)=h-1, so

\gamma(\Omega)=\frac{1}{4}((h-1)+\frac{1}{2}(h-1)^2)=\frac{1}{8}(h^2-1)

which is invariant under the Weyl group action h\rightarrow -h.

Once one has the Harish-Chandra homomorphism \gamma, for each \lambda\in\mathfrak h^* one has a homomorphism

\chi_{\lambda}: z\in Z(\mathfrak g)\rightarrow \chi_\lambda(z)=\gamma(z)(\lambda)\in \mathbf C

and the infinitesimal character of an irreducible representation of highest weight \lambda is \chi_{\lambda + \rho}.

The Casselman-Osborne Lemma

We have computed the infinitesimal character of a representation of highest weight \lambda by looking at how Z(\mathfrak g) acts on V^{\mathfrak n^+}=H^0(\mathfrak n^+,V). On V^{\mathfrak n^+},  z\in Z(\mathfrak g) acts by

z\cdot v = \chi_V(z)v

This space has weight \lambda, so U(\mathfrak h)=\mathbf C[\mathfrak h^*] acts by evaluation at \lambda

\phi\cdot v=\phi(\lambda)v

These two actions are related by the map \gamma^\prime: Z(\mathfrak g)\rightarrow U(\mathfrak h) and we have

\chi_V(z)=(\gamma^\prime(z))(\lambda)=(\gamma(z))(\lambda + \rho)

It turns out that one can consider the same question, but for the higher cohomology groups H^k(\mathfrak n^+,V). Here one again has an action of Z(\mathfrak g) and an action of U(\mathfrak h). Z(\mathfrak g) acts on k-cochains C^k(\mathfrak n^+,V)= Hom_{\mathbf C}(\Lambda^k\mathfrak n^+,V) just by acting on V, and this action commutes with d so is an action on cohomology. U(\mathfrak h) acts simultaneously on \mathfrak n^+ and on V, again in a way that descends to cohomology. The content of the Casselman-Osborne lemma is that these two actions are again related in the same way by the Harish-Chandra homomorphism. If \mu is a weight for the \mathfrak h action on H^k(\mathfrak n^+,V), then

\chi_V(z)=(\gamma^\prime(z))(\mu)=(\gamma(z))(\mu + \rho)

Since \chi_V(z)=(\gamma(z))(\lambda + \rho), one can use this equality to show that the weights occurring in H^k(\mathfrak n^+,V) must satisfy

(\mu +\rho)=w(\lambda + \rho)

and thus

\mu=w(\lambda + \rho)-\rho

for some element w\in W. Non zero elements of H^k(\mathfrak n^+,V) can be constructed with these weights, and the Casselman-Osborne lemma used to show that these are the only possible weights. This gives the computation of H^k(\mathfrak n^+,V) as an \mathfrak h – module referred to earlier in these notes, which is known as Kostant’s theorem (the algebraic proof was due to Kostant, an earlier one using geometry and sheaf cohomology was due to Bott).

For more details about this and a proof of the Casselman-Osborne lemma, see Knapp’s Lie Groups, Lie Algebras and Cohomology, where things are worked out for the case of \mathfrak g=\mathfrak{gl}(n,\mathbf C) in chapter VI.

Generalizations

So far we have been considering the case of a Cartan subalgebra \mathfrak h\subset \mathfrak g, and its orthogonal complement with a choice of splitting into two conjugate subalgebras, \mathfrak n^+ \oplus \mathfrak n^-. Equivalently, we have a choice of Borel subalgebra \mathfrak b\subset \mathfrak g, where \mathfrak b =\mathfrak h \oplus \mathfrak n^+. At the group level, this corresponds to a choice of Borel subgroup B\subset G, with the space G/B a complex projective variety known as a flag manifold. More generally, much of the same structure appears if we choose larger subgroups P \subset G containing B such that G/P is a complex projective variety of lower dimension. In these cases Lie\ P=\mathfrak l \oplus \mathfrak u^+, with \mathfrak l (the Levi subalgebra) a reductive algebra playing the role of the Cartan subalgebra, and \mathfrak u^+ playing the role of \mathfrak n^+.

In this more general setting, there is a generalization of the Harish-Chandra homomorphism, now taking Z(\mathfrak g) to Z(\mathfrak l). This acts on the cohomology groups H^k(\mathfrak u^+,V), with a generalization of the Casselman-Osborne lemma determining what representations of \mathfrak l occur in this cohomology. The Dirac cohomology formalism to be discussed later generalizes this even more, to cases of a reductive subalgebra \mathfrak r with orthogonal complement that cannot be given a complex structure and split into conjugate subalgebras. It also provides a compelling explanation for the continual appearance of \rho, as the highest weight of the spin representation.

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2 Responses to Notes on BRST VII: The Harish-Chandra Homomorphism

  1. I don’t know anything about the subject, but it sounds like a suitable title for a “The Big Bang Theory” episode.

  2. D R Lunsford says:

    This presentation of Casimirology is really first rate.

    -drl