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I've spent lots of time in Chriss and Ginzburg's "Complex Geometry and Representation Theory" and despite convolution (in Borel-Moore homology or K-theory) being very central, I feel like I'm still lacking a little understanding. I'd like some help with the following example.

On the representation theory side, I'd like to consider the following simple example. Let $G=SL_2(\mathbb{C})$, and let $T \subset B$ be the toral subgroup of diagonal matrices and the Borel subgroup consisting of upper triangular matrices respectively. Let $V_{\Lambda_1}$ denote the irreducible representation of highest weight $\Lambda_1$. Taking the tensor square of this representation yields the following decomposition into irreducibles: $V_{\Lambda_1} \otimes V_{\Lambda_1} \simeq V_{2 \Lambda_1} \oplus V_0$.

I'd like to geometrize this a la Ginzburg.

Via Borel-Weil, we know that $H^{0}(G/B, L_{\Lambda_1}) \simeq V_{\Lambda_1}$, where $L_{\Lambda_1}$ is the associated bundle $G \times_{B} \mathbb{C}^{-\Lambda_1}$. What I would like is an operation on $G$-equivariant sheaves which corresponds to the tensor product of representations, so that $H^0(G/B, L_{\Lambda_1} * L_{\Lambda_1}) \simeq V_{2 \Lambda_1} \oplus V_0$. Note that the operation cannot be the tensor product. To see this, remember that $G/B = \mathbb{P}^1$, and $L_{\Lambda_1}$ is isomorphic to $\mathscr{O}_{\mathbb{P}^1}(1)$; if I tensor this sheaf with itself and take global sections I will get the irreducible 3-dimensional representation $Sym^2(V_{\Lambda_1})=V_{2\Lambda_1}$.

Here is where I know that $*$ is supposed to be convolution, as defined by Ginzburg. (If anyone would like the definition, I can provide it, but that would lengthen this post even more).

Question 1: Is it correct to expect that $\mathscr{O}_{\mathbb{P}^1}(1) *\mathscr{O}_{\mathbb{P}^1}(1) \simeq \mathscr{O}_{\mathbb{P}^1}(2) \oplus \mathscr{O}_{\mathbb{P}^1}$? This is the only way I can see the global sections giving me the correct representation.

Question 2: If this is indeed the case, is there an explicit description in terms of global sections $T_1, T_2$ of $\mathscr{O}_{\mathbb{P}^1}(1)$, if the coordinates on $G/B=\mathbb{P}^1$ are $[T_1:T_2]$? It is easy to get the global sections $T_1^2, T_1T_2, T_2^2$ as a basis for $V_{2 \Lambda_1}$, but I cannot see how to get the basis for $\mathscr{O}_{\mathbb{P}^1}$.

It also occurs to me that I have been working with $G-$equivariant sheaves here instead of their $K$-theory, and maybe that is incorrect. I've got more thoughts, but this is already quite long for a post. Please let me know if I can provide any additional information.

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  • $\begingroup$ Here is the link to the post on Mathematics: Understanding Convolution in K-Theory via an example. (It is customary to link the copies to each other if a question is posted on several sites, see this post on meta. Although the main purpose is that people do not waste they time with writing down something which was already mentioned in the other copy - and in this case the post on Mathematics got almost no comments.) $\endgroup$ Sep 30, 2019 at 9:29

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If I understand your question correctly, you are trying to compute the convolution product on $K^G(G/B)$. Since $G/B$ is smooth, in this case the convolution product $\ast$ coincides with the tensor product $\otimes$ of $G$-equivariant sheaves (see Corollary 5.2.25. in Chriss & Ginzburg). Since $K^G(G/B)\simeq R(T)$ (Lemma 6.1.6. in Chriss & Ginzburg), the tensor product (and therefore the convolution product) of Borel-Weil line bundles with dominant weights will have the effect of adding the corresponding weights (as you computed in your question).

To get a geometric construction of $R(G)$ (the representation ring of $G$ rather than that of $T$), you might want to look at the Geometric Satake correspondence that establishes that this category is equivalent (as a tensor category) to $P_{G^\vee_\mathcal{O}}(Gr_{G^\vee})$, the category of $G^\vee_\mathcal{O}$-equivariant perverse sheaves on the affine Grassmannian $Gr_{G^\vee}$ of the Langlands dual group $G^\vee$.

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