Let $G$ be a complex semisimple Lie group. Let $\Lambda^+$ denote its dominant Weyl chamber (by fixing a Cartan and Borel) and $V_{\lambda}$ the irreducible representation of $G$ with highest weight $\lambda\in\Lambda^+$. We know that the map sending $f_{\lambda}\in End(V_{\lambda})$ to $g\in G\mapsto Tr(\rho_{\lambda}(g) f_{\lambda})$ gives rise to an isomorphism as vector spaces of $\bigoplus_{\lambda\in \Lambda^+}End(V_{\lambda})$ onto $\mathbb{C}[G]$, where $\rho_{\lambda}:G\rightarrow GL(V_{\lambda})$ is the representation map.

The ring structure on $\bigoplus_{\lambda\in \Lambda^+}End(V_{\lambda})$ transported from $\mathbb{C}[G]$ via this isomorphism is described as follows:

Let $V_{\lambda}\otimes V_{\mu}\simeq\bigoplus_{\nu\le \lambda+\mu}V^{\nu}_{\lambda,\mu}$ be the decomposition of $V_{\lambda}\otimes V_{\mu}$ into summands $V^{\nu}_{\lambda,\mu}$ which is the direct sum of a certain copies of sub-representations isomorphic to $V_{\nu}$. Now given $f_{\lambda}\in End(V_{\lambda})$ and $f_{\mu}\in End(V_{\mu})$. Then $f_{\lambda}\otimes f_{\mu}$ is in $End(V_{\lambda}\otimes V_{\mu})$. By applying a projection, we get an element of $End(V_{\lambda,\mu}^{\nu})$, and by taking a kind of trace, we get an element of $End(V_{\nu})$ which we denote by $p^{\nu}_{\lambda,\mu}(f_{\lambda}\otimes f_{\mu})$. Then $$f_{\lambda}\cdot f_{\mu}=\sum_{\nu\le \lambda+\mu}p^{\nu}_{\lambda,\mu}(f_{\lambda}\otimes f_{\mu}).$$

What I care is the following deformation of this ring structure: Choose an $X\in \mathfrak{t}$ such that $\langle X,\alpha\rangle>0$ for all positive roots $\alpha$. For each $t\in\mathbb{C}$, we define $$f_{\lambda}\cdot_t f_{\mu}:=\sum_{\nu\le \lambda+\mu}e^{t\langle X,\nu-\lambda-\mu\rangle}p^{\nu}_{\lambda,\mu}(f_{\lambda}\otimes f_{\mu}).$$ One checks easily that $\cdot_t$ defines a ring structure isomorphic to $\cdot$ for each $t$, and its limit $\cdot_{\infty}$ as $Re(t)\rightarrow +\infty$ is the one obtained from $\cdot$ by forgetting all contributions except the one by $\lambda+\mu$.

Has anyone studied $\cdot_{\infty}$, especially the geometric properties of its $Spec$? Where could I find the references?

Thank you very much.

  • $\begingroup$ After your description, isn't it just isomorphic to the polynomial ring generated by the fundamental weights in the case where $G$ is simply connected? $\endgroup$ – Wille Liou Nov 19 '19 at 9:39
  • $\begingroup$ I don't think so. Let $\lambda_1,\ldots,\lambda_r$ be the fundamental weights and $W_i=End(V_{\lambda_i})$. So you mean the ring in question is isomorphic to $Sym(W_1\oplus\cdots\oplus W_r)$. But then the summand of the latter corresponding to $\lambda=a_1\lambda_1+\cdots+a_r\lambda_r$ would be $Sym^{a_1}(W_1)\otimes\cdots\otimes Sym^{a_r}(W_r)$ which I don't believe to be $End(V_{\lambda})$. $\endgroup$ – ChiHong Chow Nov 20 '19 at 1:16
  • $\begingroup$ But $\mathrm{End}(V_{\lambda})$ is always isomorphic to $\mathbf{C}$ since $V_{\lambda}$ is irreducible, right? $\endgroup$ – Wille Liou Nov 23 '19 at 9:47
  • $\begingroup$ No here $End(V_{\lambda})$ means the space of all linear maps, equivariant or not, from $V_{\lambda}$ to itself. $\endgroup$ – ChiHong Chow Nov 25 '19 at 3:24

I believe that the structure you define is the same thing as "The Vinberg semigroup", see section 2.ii of http://front.math.ucdavis.edu/1805.07721. In particular, Spec of the $\cdot_\infty$ is isomorphic to $(G // N \times G // N_-) // T $ by Lemma 2.8 of that paper.

| cite | improve this answer | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.