Infinitesimal character - Imported Knowlpedia Content

Definition

Let $\mathfrak g$ be a complex Lie algebra and let $V$ be a module over its universal enveloping algebra. An infinitesimal character of $V$ is a unital algebra homomorphism from the center of $U(\mathfrak g)$ to $\mathbb C$ ,

\chi:Z(U(\mathfrak g))\longrightarrow\mathbb C,

such that

zv=\chi(z)v

for every $z\in Z(U(\mathfrak g))$ and $v\in V$ . A Lie-group representation or Harish–Chandra module has infinitesimal character $\chi$ when the center acts this way on its differentiated $\mathfrak g$ -module. For a real Lie algebra one uses its complexification.

Harish–Chandra parameters

When $\mathfrak g$ is semisimple and $\mathfrak h$ is a Cartan subalgebra, the Harish–Chandra isomorphism identifies infinitesimal characters with $W$ -orbits in $\mathfrak h^*$ . With the shifted convention, a highest-weight module of highest weight $\lambda$ has the character obtained by evaluating Weyl-invariant polynomials at $\lambda+\rho$ . Consequently, distinct representations can have the same infinitesimal character.

Existence and generalized characters

An arbitrary module need not have an infinitesimal character: central elements may act non-scalarly. Irreducible modules in the usual complex semisimple settings do have one under the relevant form of Schur's lemma. Finite-length modules are sometimes allowed a generalized infinitesimal character, meaning that a power of each $z-\chi(z)$ acts as zero; this is weaker than scalar central action.

Relation to global characters

If an admissible representation has infinitesimal character $\chi$ , its global character is a joint eigendistribution for the invariant differential operators corresponding to the center, with eigenvalues prescribed by $\chi$ . Infinitesimal and global characters are therefore related but are not the same kind of object.

References

Anthony W. Knapp, Representation Theory of Semisimple Groups: An Overview Based on Examples, Princeton University Press, 1986; reprint 2001. Author-maintained record. Relevant: Chapter VIII on infinitesimal characters.
Jacques Dixmier, Enveloping Algebras, Graduate Studies in Mathematics 11, American Mathematical Society, 1996. DOI record. Relevant: Chapter 7 on central characters.