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First, we may use the based cofiber sequence $ S^{2n-1}_{+}\to D^{2n}_{+} \to S^{2n}_{+} $ and its associated Gysin sequence $$0 \leftarrow K_{A\times \mathbb{T}}(S^{2n-1}) \leftarrow R(A)[z^{\pm1}] \xleftarrow{\chi} R(A)[z^{\pm1}] \leftarrow \cdots $$ If we write $$R(A) = \mathbb{Z}[t_{1}^{\pm1},\cdots,t_{n}^{\pm1}]/(1-t_{1}\cdots t_{n})$$ in this case $\chi=\prod_{i=1}^{n}(1-t_{i}z)$ and we can obtain $$K_{A}(\mathbb{C}P^{n-1})= R(A)[z^{\pm1}]/(\prod_{i=1}^{n}(1-t_{i}z)) = \mathbb{Z}[t_{1}^{\pm1},\cdots,t_{n}^{\pm1},z^{\pm1}]/(1-t_{1}\cdots t_{n},\prod_{i=1}^{n}(1-t_{i}z)).$$ This is the method discussed in Section 10 of Greenlees, J. P. C. "Equivariant formal group laws and complex oriented cohomology theories." (https://projecteuclid.org/journals/homology-homotopy-and-applications/volume-3/issue-2/Equivariant-formal-group-laws-and-complex-oriented-cohomology-theories/hha/1139840255.full)

The two later descriptions should be isomorphic by looking at the following embedding $$R(A)[z^{\pm1}]/\prod_{i=1}^{n}(1-t_{i}z) \to \prod_{i=1}^{n}R(A)[z^{\pm1}]/(1-t_{i}z)\cong \text{Map}([n],R(A)), \overline{f}(z) \mapsto (f(t_{1}^{-1}),\cdots,f(t_{n}^{-1})) $$ and checking that the image is precisely the elements given by the GKM description (which should be true, although I haven't checked it carefully).

First, we may use the based cofiber sequence $S^{2n-1}_{+}\to D^{2n}_{+} \to S^{2n}_{+}$ and its associated Gysin sequence $$0 \leftarrow K_{A\times \mathbb{T}}(S^{2n-1}) \leftarrow R(A)[z^{\pm1}] \xleftarrow{\chi} R(A)[z^{\pm1}] \leftarrow \cdots $$ If we write $$R(A) = \mathbb{Z}[t_{1}^{\pm1},\cdots,t_{n}^{\pm1}]/(1-t_{1}\cdots t_{n})$$ in this case $\chi=\prod_{i=1}^{n}(1-t_{i}z)$ and we can obtain $$K_{A}(\mathbb{C}P^{n-1})= R(A)[z^{\pm1}]/(\prod_{i=1}^{n}(1-t_{i}z)) = \mathbb{Z}[t_{1}^{\pm1},\cdots,t_{n}^{\pm1},z^{\pm1}]/(1-t_{1}\cdots t_{n},\prod_{i=1}^{n}(1-t_{i}z)).$$ This is the method discussed in Section 10 of Greenlees, J. P. C. "Equivariant formal group laws and complex oriented cohomology theories." (https://projecteuclid.org/journals/homology-homotopy-and-applications/volume-3/issue-2/Equivariant-formal-group-laws-and-complex-oriented-cohomology-theories/hha/1139840255.full)

The two later descriptions should be isomorphic by looking at the following embedding $$R(A)[z^{\pm1}]/\prod_{i=1}^{n}(1-t_{i}z) \to \prod_{i=1}^{n}R(A)[z^{\pm1}]/(1-t_{i}z)\cong \text{Map}([n],R(A)) $$ given by $$\overline{f}(z) \mapsto (f(t_{1}^{-1}),\cdots,f(t_{n}^{-1}))$$ and checking that the image is precisely the elements given by the GKM description (which should be true, although I haven't checked it carefully).

The first way (which can be wrong and please correct me if there is any mistake) is to use the observation that $$K_{T}(G/H)=K_{G}(G\times_{T}G/H)\cong K_{G}(G/T\times G/H)$$ where the second isomorphism is due to the fact there is a $G$-equivariant isomorphism $$G\times_{T}G/H\to G/T\times G/H, ~[g_1,g_2H]_{T} \mapsto (g_{1}T,g_1g_2H).$$ Applying Hogkins' spectral sequence (see, e.g., Theorem 2.3 in Brylinski, Jean-Luc, and Bin Zhang. "Equivariant K-Theory of Simply Connected Lie Groups." https://arxiv.org/abs/dg-ga/9710035) we can obtain $$K_{T}(G/H)\cong K_{G}(G/T)\otimes_{R(G)}K_{G}(G/H)\cong R(T)\otimes_{R(G)}R(H)$$ since the spectral sequence collapses.

My questions is, is there any mistake in my first way of describing equivariant $K$-theory? If not, how to find explicit isomorphism between the first and the second description? I am also not sure if I miss any important references (I just jumped in this this question so it is very likely I missed many fundamental results) so if anyone can give me suggestions on useful references I would really appreciate it.

The first way (which can be wrong and please correct me if there is any mistake) is to use the observation that $$K_{T}(G/H)=K_{G}(G\times_{T}G/H)\cong K_{G}(G/T\times G/H)$$ where the second isomorphism is due to the fact there is a $G$-equivariant isomorphism $$G\times_{T}G/H\to G/T\times G/H, ~[g_1,g_2H]_{T} \mapsto (g_{1}T,g_1g_2H)$$ see, e.g., Chapter 3 Example 4.4 in Anderson, David, and William Fulton, "Equivariant cohomology in algebraic geometry."

Applying Hogkins' spectral sequence (see, e.g., Theorem 2.3 in Brylinski, Jean-Luc, and Bin Zhang. "Equivariant K-Theory of Simply Connected Lie Groups." https://arxiv.org/abs/dg-ga/9710035) we can obtain $$K_{T}(G/H)\cong K_{G}(G/T)\otimes_{R(G)}K_{G}(G/H)\cong R(T)\otimes_{R(G)}R(H)$$ since the spectral sequence collapses.

My questions is, is there any mistake in my first way of describing equivariant $K$-theory? If not, how to find explicit isomorphism between the first and the second description? I am also not sure if I miss any important references (I just jumped in this this question so it is very likely I missed many fundamental results, and I am also aware of the fact that the references I mentioned here are neither the earliest ones nor the most complete ones but I referred to them as explicit descriptions are given) so if anyone can give me suggestions on useful references I would really appreciate it.