Convolution in K-Theory via an Example (From StackExchange)

Question

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.

Answer 1

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 $G^\vee$.