# Understanding Convolution in K-Theory via an example

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 $$\mathscr{O}_{\mathbb{P}^1}$$.

I've got more thoughts, but I think this is already quite long for a post. Please let me know if I can provide any additional information.