Let $G = SL(2,\mathbb C)$, which is an algebraic group of type $A_1$ over $\mathbb C$. Let $B$ be a Borel subgroup of $G$. Let $X =G/B$. Then $X \cong \mathbb P^1$ and it has a stratification $X = X_{e} \sqcup X_{s}$, where $X_e = $pt and $X_s \cong \mathbb A_1$. Let $j_s: X_s \hookrightarrow X$ be the inclusion. In the following, for any space $Y$, $\underline{\mathbb C}_Y$ denotes the constant sheaf on $Y$ with value $\mathbb C$.
I have to show
$j_{s!}\underline{\mathbb C}_{X_s}*\underline{\mathbb C}_{\mathbb P^1} = \underline{\mathbb C}_{\mathbb P^1}[-2]$.
Already know: for $a: X \rightarrow $pt, $j_{s!}\underline{\mathbb C}_{X_s}*\underline{\mathbb C}_{\mathbb P^1} = a^*a_!j_{s!}\underline{\mathbb C}_{X_s}$. But what I cannot go through is
- $a_!j_{s!}\underline{\mathbb C}_{X_s} = \underline{\mathbb C}_{\text{pt}}[-2] = H_c^{\bullet}(\mathbb A^1)$.
- $a^*(\underline{\mathbb C}_{\text{pt}}[-2])$ (or $a^*(H_c^{\bullet}(\mathbb A^1))$) $=\underline{\mathbb C}_{\mathbb P^1}[-2]$.
Would anyone please explain me the equations, or give another proof, other than using the isomorphism $K(D^b_B(X)) \cong H_{q=-1}$, where $K(D^b_B(X))$ is the Grothendieck group of $D^b_B(X)$ with the convolution product and $H_{q=-1}$ is the Hecke algebra specializing at $q =-1$?
For the definition of convolution product, please read Perverse sheaves on affine Grassmannians and Langlands duality or other references.
Thanks.