# What's the relation of the Hecke algebra of a pair and the flag variety?

Let $G$ be a real semisimple Lie group and $K$ a maximal compact subgroup. Let $\mathfrak{g}$ and $\mathfrak{k}$ be the complexified Lie algebra of $G$ and $K$, respectively.

Then the Hecke algebra of the pair $(\mathfrak{g},K)$, denoted by $R(\mathfrak{g}, K)$, is the algebra of left $K$ finite distributions on $G$ with support in $K$, with convolution as multiplication. (This algebra is not exactly the same as the Iwahori–Hecke algebra and I'm not clear about their relation.)

Algebraically, Let $U(\mathfrak{g})$ and $U(\mathfrak{k})$ be the universal enveloping algebras of $\mathfrak{g}$ and $\mathfrak{k}$ respectively. Let $R(K)$ be the algebra of left $K$ finite distributions on $K$ with convolution as multiplication. Then the Hecke algebra $R(\mathfrak{g}, K)$ is the smash product $R(K)\sharp U(\mathfrak{g})$ then quotient by the ideal generated by $U(\mathfrak{k})$.

An important property of Hecke algebra $R(\mathfrak{g}, K)$ is the following

Theorem: There is an equivalence between the category of $(\mathfrak{g}, K)$-modules and the category of approximately unital left $R(\mathfrak{g}, K)$-modules.

See "Cohomological Induction and Unitary Representations" by Knapp and Vogan, Section 1.4 and 1.6 for details.

Now it is well-known in geometric representation theory that $(\mathfrak{g}, K)$-modules with a fixed infinitesimal character is equivalent to twisted D-modules on the flag variety of $\mathfrak{g}$.

My question is: could we put these two ways together? In other words, could we realize the algebra $R(\mathfrak{g}, K)$ as some geometric object on the flag variety of $\mathfrak{g}$?