# Tensor product of irreducible ''anti-dominant'' representations

Let us consider the algebra $\mathfrak{gl}_{\infty}$ (or $\mathfrak{gl}_n$, or just any finite-dimensional semisimple Lie algebra; howerer, I am primarily concerned with the case of $\mathfrak{gl}_{\infty}$ and $\mathfrak{gl}_n$). I call its highest weight representation $V$ of weight $\lambda$ anti-dominant if $\lambda=\sum -k_i\omega_i$ where $\omega_i=(\dots 1\ 1\ 0\ 0\dots)$ is the $i$-th fundamental weight, $k_i\in\mathbb{Z}_{\geq 0}$ and only a finite number of them is nonzero.

Is it true that the tensor product of two irreducible anti-dominant representaions is semisimple? Or maybe at least the sub-representation generated by the highest weight vector of the tensor product is irreducible? How can one prove (disprove) this?

• You are looking at the tensor product of two infinite dimensional representations, which falls outside the usual framework of representaton theory. Possibly something can be said, but for example you'd want to be more explicit about such irreducible anti-dominant representations: these include the simple Verma modules. What can be said for example about the tensor product of such a module with itself? – Jim Humphreys May 26 '18 at 22:27
• After some consideration, I can show that $L(-k\omega_i)\subset L(-\omega_i)^{\otimes k}$ (where $L$ denotes irreducible modules): one has to construct a certain anti-linear anti-involution of the algebra and a contravariant positive-definite Hermitian form on $L(-\omega_i)$. However, this anti-involution depends on $i$, which makes it impossible to use it for the general case. – Gregg May 29 '18 at 20:46
• As for your question, it is easy to show that if weights $\lambda$ and $\mu$ are “anti-dominant“ (in my sense) and the Verma module $M(\lambda+\mu)$ is simple (so $\lambda+\mu$ is anti-dominant in the usual sense), then the conjectured statement (obviously) holds. – Gregg May 29 '18 at 20:47
• I'm still unclear about what your main question is (it should be highlighted). While it may be true that your follow-up question is true, it's very unlikely that the initial question about semisimpllicity is true. (Would this have interesting impllications?) I suspect any tensor product of antidominant Verma modules (i.e., simple ones) would be a counterexample. – Jim Humphreys Jun 3 '18 at 12:38
• Sorry, now it should be clear from the question that I am mostly interested in the second question. However, my remark in the comments about $L(-k\omega_i)$ is to the first question. The second remark is, of course, to the second (main) question. – Gregg Jun 4 '18 at 13:29