# Weights of a locally finite-dimensional module

Let $\frak g$ be a complex finite-dimensional simple Lie algebra and $V$ be a $\frak g$-module with weights bounded by above by some fixed weight and suppose that $V$ is locally finite-dimensional. How to show that the set of weights of $V$ is finite?

-

I'm not sure about the concept of weights for an arbitrary Lie algebra $\mathfrak g$, so let me assume that $\mathfrak g$ is semisimple. [Added: Since this was written, the OP has edited the question to specify that $\mathfrak g$ is in fact simple; so this hypothesis is fine.]

Then any finite-dimensional $\mathfrak g$ rep. is the direct sum of simple ones. Since $V$ is locally finite-dimensional, it is the direct limit of its finite-dimensional subreps., and so is itself a direct sum of simple reps., say $V = \bigoplus_{i} V_{\lambda_i},$ where $V_{\lambda_i}$ is a highest weight rep. of weight $i$. Since the weights of $V$ are bounded above, the $\lambda_i$ are bounded, and hence (although the index set of $i$ may be infinite) the $\lambda_i$ range over a finite set of weights. Thus the weights of the $V_{\lambda_i}$ range over a finite set of weights, and hence so do those of $V$.

-