Unitary Highest Weight Representations

Inner Products on Highest Weight Representations

Unitary Highest Weight Representation

We call a highest weight representation for of weight unitary if there exists a positive definite contravariant form (i.e., a complex inner product) on . In this case, we call a unitary highest weight representation, in abbreviation UHW representation.

Lemma

Suppose is a -UHW representation for , and is a submodule of it, then the orthogonal complement is also a submodule, and both and are the direct sum of finite dimensional -eigenspaces.

Proof For any , and , we have so . Hence, is a submodule. Since is diagonalizable on , is the direct sum of finite dimensional spaces , for being eigenspaces of .

Proposition

If is a pointed UHW representation of weight , then the submodule is irreducible and isomorphic to .

Proof In the category , is the initial object. So there exists . Notice that so . Moreover, suppose is a proper submodule, then , so . Hence, , thus It follows that , so is irreducible. As is the only irreducible quotient of , we have .

A direct consequence of the above proposition is that the contravariant form on can be restricted to :

Corollary

If is a UHW representation for , then restricts to a positive multiple of the contravariant form on .

Corollary

Any -UHW representation is completely reducible. Moreover, if is the representation of induced by , then it is completely reducible as well.

Proof We first show that can be decomposed into irreducible modules of through a recursive process, and then the same process will work for . From the above proposition, we know that is a irreducible submodule for being the highest weight vector. can be decomposed into direct sum of irreducible submodules by the following procedure defined recursively:

1. Pick some highest weight , then is an irreducible submodule. Let , which is also a submodule.
2. If , then stop. Otherwise, apply the same procedure on .

This algorithm must terminate at each energy level, because each -eigenspaces is finite dimensional, and taking the orthogonal sum over all levels (countably many) exhausts .

Quantization

Friedan–Qiu–Shenker Classification

An -highest weight representation for is unitary if and only if either and or for some integers , . In particular, and .

Proof See Ref.1.

Energy Bounds

Lemma

Fix a UHW representation of . Suppose containing some vector such that is an irreducible HW representation of , and there exists such that . Then for all , Moreover,

Proof We use the standard basis of . Fix some , since is an irreducible HW representation of , we can write Let be the highest weight of on . Then applying the commutation relation gives us for some . Note that the Casimir operator lies in the center of , so it acts as a scalar multiplication. Note that for any highest weight vector of , thus acts as multiplying by on the whole , in particular, . Therefore, substituting into this equation gives us it follows that Therefore, As , it further deduces to Now since , we have . By our assumption that is an eigenvector of , we have , hence, Now, notice that a highest weight vector is also a vector in , so is in some -eigenspace for , that is , so Subtracting this from , we get Since and , the above equation implies that Similarly, adding and , As a result, By the commutation relation , one can derive that Together with , this gives the basic estimate for all . To show the second estimate, we first observe that if , then implies that , as is nonnegative, , so . So we may assume without loss of generality that . Note that so by functional calculus, . Hence, Case , then , so . Case , then . Substituting these bounds into the initial inequality yields the desired estimate for both cases, completing the proof.

Remark

The above proof closely follows the original argument of Goodman and Wallach (1985). However, it is worth noting that there appears to be a minor typo in their equation (2.8): a term of the form seems to be missing. This omission has been corrected in the version presented here. The discrepancy is ultimately harmless for our purposes, since we only require the expression to be bounded by a polynomial, and the conclusion remains unaffected.

Lemma

Fix a UHW representation of . For all and ,

Proof Every in the Witt is a finite linear combination of generators , so the triangle inequality guarantees that the inequality holds for all . Furthermore, is self-adjoint with discrete spectrum, so any can be decomposed into eigenvectors of , and we have proved the desired inequality for each eigenvector, so it holds for all .

Theorem

If is a UHW representation of , then it can be extended to a continuous projective representation of , such that for all and , where is the completion of with respect to , and are scales of associated with . In particular, if and , then is essentially self-adjoint as a unbounded operator on .

Proof By the above lemma, since is bounded, it can be extended to . Fix a UHW representation for of weight . let , then is a Hilbert space with inner product From the above estimate, we know that Recall that (ref. proposition), so . Note that so as well. In particular, if , then thus , is formally Hermitian, so it is essentially self-adjoint by Nelson’s commutator theorem.

Stone’s theorem immediately implies the following corollary:

Corollary

Suppose is a real vector field on . Then is a strongly continuous one-parameter group of unitary operators on .

References

Details of the non-unitarity proof for highest weight representations of the Virasoro algebra

Goodman and Wallach, Projective unitary positive-energy representations of