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Let $G$ be a simply connected simple algebraic group over $\mathbb{C}$ and $\mathfrak{g}$ be its Lie algebra. Let $\mathcal{N}$ be the nilpotent cone of $\mathfrak{g}$ consists of all nilpotent elements. Let $\mathscr{B}$ be the flag variety of $G$ consists of all borel subalgebras of $\mathfrak{g}$. Let $Z:=\{(\mathfrak{b},\mathfrak{b}',n)\mid \mathfrak{b},\mathfrak{b}'\in\mathscr{B},n\in\mathcal{N}, n\in\mathfrak{b}\cap\mathfrak{b}'\}$ be the Steinberg variety and $\mu:Z\to\mathcal{N}$ be the canonical projection. Let $\mathcal{O}$ be a nilpotent orbit and $\overline{\mathcal{O}}$ be its Zariski closure. Let $\tilde{\mathbf{H}}$ be the affine Hecke algebra associated with $G$, i.e., $\tilde{\mathbf{H}}\simeq H\ltimes X$, where $H$ is the finite Hecke algebra and $X$ be the weight lattice of $G$.

Due to the work of Kazhdan-LusztigKazhdan–Lusztig and Ginzburg, there exists two equivariant K-theory realization of $\tilde{\mathbf{H}}$. The first one is realization via topological K-theory, that is $$K_{0,top}^{G\times\mathbb{C}^*}(Z)\simeq\tilde{\mathbf{H}},$$$$K_{0,\text{top}}^{G\times\mathbb{C}^*}(Z)\simeq\tilde{\mathbf{H}},$$ where $K_{0,top}(-)$$K_{0,\text{top}}(-)$ is the 0$0$-th topological K-theory. See [1] Kazhdan, D. Kazhdan & G. Lusztig, G.: Proof of the Deligne-Langlands Conjecture for Hecke AlgebrasProof of the Deligne–Langlands Conjecture for Hecke Algebras. Invent. math. 87, 153-215  (1987), 153–215.

The other is realization via equivariant algebraic K-theory by V. Ginzburg, that is $$K_{0}^{G\times\mathbb{C}^*}(Z)\simeq\tilde{\mathbf{H}},$$ where $K_0$ is the 0$0$-th algebraic K-theory. See [2] V. Ginzburg, V.: "Lagrangian" Construction for Representations of Hecke Algebras"Lagrangian" Construction for Representations of Hecke Algebras. Adv. math. 63. 100-112  (1987), 100–112.

Let $Z_{\overline{\mathcal{O}}}:=\mu^{-1}(\overline{\mathcal{O}})$, and let $i:Z_{\overline{\mathcal{O}}}\hookrightarrow Z$. Then $i_*: K_{0,top}^{G\times\mathbb{C}^*}(Z_{\overline{\mathcal{O}}})\to K_{0,top}^{G\times\mathbb{C}^*}(Z)$$i_*: K_{0,\text{top}}^{G\times\mathbb{C}^*}(Z_{\overline{\mathcal{O}}})\to K_{0,\text{top}}^{G\times\mathbb{C}^*}(Z)$ is a injective map. In [1], Kazhdan-LusztigKazhdan–Lusztig proved it by showshowing $K^{G\times\mathbb{C}^*}_{1,top}(Z_{\mathcal{O}})=0$$K^{G\times\mathbb{C}^*}_{1,\text{top}}(Z_{\mathcal{O}})=0$, where $Z_{\mathcal{O}}:=\mu^{-1}(\mathcal{O})$. This is also true for algebraic K-theory, that is $i_*:K_0^{G\times\mathbb{C}^*}(Z_{\overline{\mathcal{O}}})\to K_0^{G\times\mathbb{C}^*}(Z)$ being injective (see [3, Proposition 3.3] Tanisaki, T. and Xi, N.: Kazhdan-Lusztig Basis and A Geometric Filtration of an affine Hecke AlgebraKazhdan–Lusztig Basis and A Geometric Filtration of an affine Hecke Algebra, Nagoya Math. J. 182 (2006), 285-311285–311).

For $Z$ and $Z_{\overline{\mathcal{O}}}$, the 0$0$-th algebraic K-theory and 0$0$-th topological K-theory coincide. But it is not evident that the 1$1$-thst K-theorystheories of $Z_{\mathcal{O}}$ coincide. In fact, Xi conjectedconjectured that $K_{1}^{G\times\mathbb{C}^*}(Z_{\mathcal{O}})=0$, where $K_1$ is the 1$1$-thst algebraic K-theory.

My question: How to prove $i_*:K_0^{G\times\mathbb{C}^*}(Z_{\overline{\mathcal{O}}})\to K_0^{G\times\mathbb{C}^*}(Z)$ is injective, if we only use algebraic K-theory?

Let $G$ be a simply connected simple algebraic group over $\mathbb{C}$ and $\mathfrak{g}$ be its Lie algebra. Let $\mathcal{N}$ be the nilpotent cone of $\mathfrak{g}$ consists of all nilpotent elements. Let $\mathscr{B}$ be the flag variety of $G$ consists of all borel subalgebras of $\mathfrak{g}$. Let $Z:=\{(\mathfrak{b},\mathfrak{b}',n)\mid \mathfrak{b},\mathfrak{b}'\in\mathscr{B},n\in\mathcal{N}, n\in\mathfrak{b}\cap\mathfrak{b}'\}$ be the Steinberg variety and $\mu:Z\to\mathcal{N}$ be the canonical projection. Let $\mathcal{O}$ be a nilpotent orbit and $\overline{\mathcal{O}}$ be its Zariski closure. Let $\tilde{\mathbf{H}}$ be the affine Hecke algebra associated with $G$, i.e., $\tilde{\mathbf{H}}\simeq H\ltimes X$, where $H$ is the finite Hecke algebra and $X$ be the weight lattice of $G$.

Due to the work of Kazhdan-Lusztig and Ginzburg, there exists two equivariant K-theory realization of $\tilde{\mathbf{H}}$. The first one is realization via topological K-theory, that is $$K_{0,top}^{G\times\mathbb{C}^*}(Z)\simeq\tilde{\mathbf{H}},$$ where $K_{0,top}(-)$ is the 0-th topological K-theory. See [1] D. Kazhdan & G. Lusztig.: Proof of the Deligne-Langlands Conjecture for Hecke Algebras. Invent. math 87, 153-215(1987).

The other is realization via equivariant algebraic K-theory by V. Ginzburg, that is $$K_{0}^{G\times\mathbb{C}^*}(Z)\simeq\tilde{\mathbf{H}},$$ where $K_0$ is the 0-th algebraic K-theory. See [2] V. Ginzburg.: "Lagrangian" Construction for Representations of Hecke Algebras. Adv. math. 63. 100-112(1987).

Let $Z_{\overline{\mathcal{O}}}:=\mu^{-1}(\overline{\mathcal{O}})$, and let $i:Z_{\overline{\mathcal{O}}}\hookrightarrow Z$. Then $i_*: K_{0,top}^{G\times\mathbb{C}^*}(Z_{\overline{\mathcal{O}}})\to K_{0,top}^{G\times\mathbb{C}^*}(Z)$ is a injective map. In [1], Kazhdan-Lusztig proved it by show $K^{G\times\mathbb{C}^*}_{1,top}(Z_{\mathcal{O}})=0$, where $Z_{\mathcal{O}}:=\mu^{-1}(\mathcal{O})$. This is also true for algebraic K-theory, that is $i_*:K_0^{G\times\mathbb{C}^*}(Z_{\overline{\mathcal{O}}})\to K_0^{G\times\mathbb{C}^*}(Z)$ being injective (see [3, Proposition 3.3] Tanisaki, T. and Xi, N.: Kazhdan-Lusztig Basis and A Geometric Filtration of an affine Hecke Algebra, Nagoya Math. J. 182 (2006), 285-311).

For $Z$ and $Z_{\overline{\mathcal{O}}}$, the 0-th algebraic K-theory and 0-th topological K-theory coincide. But it is not evident that the 1-th K-theorys of $Z_{\mathcal{O}}$ coincide. In fact, Xi conjected that $K_{1}^{G\times\mathbb{C}^*}(Z_{\mathcal{O}})=0$, where $K_1$ is the 1-th algebraic K-theory.

My question: How to prove $i_*:K_0^{G\times\mathbb{C}^*}(Z_{\overline{\mathcal{O}}})\to K_0^{G\times\mathbb{C}^*}(Z)$ is injective, if we only use algebraic K-theory?

Let $G$ be a simply connected simple algebraic group over $\mathbb{C}$ and $\mathfrak{g}$ be its Lie algebra. Let $\mathcal{N}$ be the nilpotent cone of $\mathfrak{g}$ consists of all nilpotent elements. Let $\mathscr{B}$ be the flag variety of $G$ consists of all borel subalgebras of $\mathfrak{g}$. Let $Z:=\{(\mathfrak{b},\mathfrak{b}',n)\mid \mathfrak{b},\mathfrak{b}'\in\mathscr{B},n\in\mathcal{N}, n\in\mathfrak{b}\cap\mathfrak{b}'\}$ be the Steinberg variety and $\mu:Z\to\mathcal{N}$ be the canonical projection. Let $\mathcal{O}$ be a nilpotent orbit and $\overline{\mathcal{O}}$ be its Zariski closure. Let $\tilde{\mathbf{H}}$ be the affine Hecke algebra associated with $G$, i.e., $\tilde{\mathbf{H}}\simeq H\ltimes X$, where $H$ is the finite Hecke algebra and $X$ be the weight lattice of $G$.

Due to the work of Kazhdan–Lusztig and Ginzburg, there exists two equivariant K-theory realization of $\tilde{\mathbf{H}}$. The first one is realization via topological K-theory, that is $$K_{0,\text{top}}^{G\times\mathbb{C}^*}(Z)\simeq\tilde{\mathbf{H}},$$ where $K_{0,\text{top}}(-)$ is the $0$-th topological K-theory. See [1] Kazhdan, D. & Lusztig, G.: Proof of the Deligne–Langlands Conjecture for Hecke Algebras. Invent. math. 87  (1987), 153–215.

The other is realization via equivariant algebraic K-theory by V. Ginzburg, that is $$K_{0}^{G\times\mathbb{C}^*}(Z)\simeq\tilde{\mathbf{H}},$$ where $K_0$ is the $0$-th algebraic K-theory. See [2] Ginzburg, V.: "Lagrangian" Construction for Representations of Hecke Algebras. Adv. math. 63  (1987), 100–112.

Let $Z_{\overline{\mathcal{O}}}:=\mu^{-1}(\overline{\mathcal{O}})$, and let $i:Z_{\overline{\mathcal{O}}}\hookrightarrow Z$. Then $i_*: K_{0,\text{top}}^{G\times\mathbb{C}^*}(Z_{\overline{\mathcal{O}}})\to K_{0,\text{top}}^{G\times\mathbb{C}^*}(Z)$ is a injective map. In [1], Kazhdan–Lusztig proved it by showing $K^{G\times\mathbb{C}^*}_{1,\text{top}}(Z_{\mathcal{O}})=0$, where $Z_{\mathcal{O}}:=\mu^{-1}(\mathcal{O})$. This is also true for algebraic K-theory, that is $i_*:K_0^{G\times\mathbb{C}^*}(Z_{\overline{\mathcal{O}}})\to K_0^{G\times\mathbb{C}^*}(Z)$ being injective (see [3, Proposition 3.3] Tanisaki, T. and Xi, N.: Kazhdan–Lusztig Basis and A Geometric Filtration of an affine Hecke Algebra, Nagoya Math. J. 182 (2006), 285–311).

For $Z$ and $Z_{\overline{\mathcal{O}}}$, the $0$-th algebraic K-theory and $0$-th topological K-theory coincide. But it is not evident that the $1$-st K-theories of $Z_{\mathcal{O}}$ coincide. In fact, Xi conjectured that $K_{1}^{G\times\mathbb{C}^*}(Z_{\mathcal{O}})=0$, where $K_1$ is the $1$-st algebraic K-theory.

My question: How to prove $i_*:K_0^{G\times\mathbb{C}^*}(Z_{\overline{\mathcal{O}}})\to K_0^{G\times\mathbb{C}^*}(Z)$ is injective, if we only use algebraic K-theory?

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An injective map in equivariant algebraic K-theory

Let $G$ be a simply connected simple algebraic group over $\mathbb{C}$ and $\mathfrak{g}$ be its Lie algebra. Let $\mathcal{N}$ be the nilpotent cone of $\mathfrak{g}$ consists of all nilpotent elements. Let $\mathscr{B}$ be the flag variety of $G$ consists of all borel subalgebras of $\mathfrak{g}$. Let $Z:=\{(\mathfrak{b},\mathfrak{b}',n)\mid \mathfrak{b},\mathfrak{b}'\in\mathscr{B},n\in\mathcal{N}, n\in\mathfrak{b}\cap\mathfrak{b}'\}$ be the Steinberg variety and $\mu:Z\to\mathcal{N}$ be the canonical projection. Let $\mathcal{O}$ be a nilpotent orbit and $\overline{\mathcal{O}}$ be its Zariski closure. Let $\tilde{\mathbf{H}}$ be the affine Hecke algebra associated with $G$, i.e., $\tilde{\mathbf{H}}\simeq H\ltimes X$, where $H$ is the finite Hecke algebra and $X$ be the weight lattice of $G$.

Due to the work of Kazhdan-Lusztig and Ginzburg, there exists two equivariant K-theory realization of $\tilde{\mathbf{H}}$. The first one is realization via topological K-theory, that is $$K_{0,top}^{G\times\mathbb{C}^*}(Z)\simeq\tilde{\mathbf{H}},$$ where $K_{0,top}(-)$ is the 0-th topological K-theory. See [1] D. Kazhdan & G. Lusztig.: Proof of the Deligne-Langlands Conjecture for Hecke Algebras. Invent. math 87, 153-215(1987).

The other is realization via equivariant algebraic K-theory by V. Ginzburg, that is $$K_{0}^{G\times\mathbb{C}^*}(Z)\simeq\tilde{\mathbf{H}},$$ where $K_0$ is the 0-th algebraic K-theory. See [2] V. Ginzburg.: "Lagrangian" Construction for Representations of Hecke Algebras. Adv. math. 63. 100-112(1987).

Let $Z_{\overline{\mathcal{O}}}:=\mu^{-1}(\overline{\mathcal{O}})$, and let $i:Z_{\overline{\mathcal{O}}}\hookrightarrow Z$. Then $i_*: K_{0,top}^{G\times\mathbb{C}^*}(Z_{\overline{\mathcal{O}}})\to K_{0,top}^{G\times\mathbb{C}^*}(Z)$ is a injective map. In [1], Kazhdan-Lusztig proved it by show $K^{G\times\mathbb{C}^*}_{1,top}(Z_{\mathcal{O}})=0$, where $Z_{\mathcal{O}}:=\mu^{-1}(\mathcal{O})$. This is also true for algebraic K-theory, that is $i_*:K_0^{G\times\mathbb{C}^*}(Z_{\overline{\mathcal{O}}})\to K_0^{G\times\mathbb{C}^*}(Z)$ being injective (see [3, Proposition 3.3] Tanisaki, T. and Xi, N.: Kazhdan-Lusztig Basis and A Geometric Filtration of an affine Hecke Algebra, Nagoya Math. J. 182 (2006), 285-311).

For $Z$ and $Z_{\overline{\mathcal{O}}}$, the 0-th algebraic K-theory and 0-th topological K-theory coincide. But it is not evident that the 1-th K-theorys of $Z_{\mathcal{O}}$ coincide. In fact, Xi conjected that $K_{1}^{G\times\mathbb{C}^*}(Z_{\mathcal{O}})=0$, where $K_1$ is the 1-th algebraic K-theory.

My question: How to prove $i_*:K_0^{G\times\mathbb{C}^*}(Z_{\overline{\mathcal{O}}})\to K_0^{G\times\mathbb{C}^*}(Z)$ is injective, if we only use algebraic K-theory?