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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.

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  • $\begingroup$ 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
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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.

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