I am looking at references for computing $$K_{T}(G/H)$$ where $G$ is a compact connected Lie group with maximal torus $T$, and $H\subset G$ is a corank one Lie subgroup such that $G/H\cong S^{2k-1}$ is an odd dimensional sphere. The $T$-action on the homogeneous space $G/H$ is the restriction of canonical left $G$-action on $G/H$. The examples I am particular interested in are the cases when $(G=\text{U}(n), H = \text{U}(n-1))$, $(G=\text{SU}(n), H = \text{SU}(n-1))$ or $(G=\text{Sp}(n), H = \text{Sp}(n-1))$.

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.

To compare this result with the second method I will mention later, let's focus on the case when $(G=\text{U}(n), H = \text{U}(n-1))$. In this case the explicit formula is given by \begin{equation}\label{eq:1} K_{T}(\text{U}(n)/\text{U}(n-1))=\mathbb{Z}[z_{1}^{\pm 1},\cdots,z_{n}^{\pm1}]/\prod_{i=1}^{n}(1-z_{i}). \end{equation} On the other hand, in this case we notice $G/H\cong S^{2n-1}$ and the $T$-action is the restriction of the standard $T$-action on $\mathbb{C}^{n}$ to the unit sphere. In particular there is a copy of $\mathbb{T}=S^{1} \subset T$ that acts freely (the diagonal one), thus we can decompose $T=A\times\mathbb{T}$ where $A=T\cap \text{SU}(n)$ is of rank $n-1$, and it follows that $$K_{T}(S^{2n-1})=K_{A\times \mathbb{T}}(S^{2n-1})\cong K_{A}(\mathbb{C}P^{n-1}).$$ There are two ways to write $K_{A}(\mathbb{C}P^{n-1})$.

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)

Alternatively, since the $A$-action on $M=\mathbb{C}P^{n-1}$ is GKM (i.e., $M^{A}$ is finite and for every codimension 1 subtorus $A'$ of $A$ the components of $M^{A'}$ are of dimension at most 2), there is a GKM description of $K_{A}(M)$ as follows (see, e.g., Guillemin, Victor, Silvia Sabatini, and Catalin Zara. "Equivariant K-theory of GKM bundles." https://link.springer.com/article/10.1007/s10455-012-9331-3).

If we identify $M^{A}=[n]=\{1,\cdots,n\}$, we can write $$K_{A}(M)=\{f\in \text{Map}([n],R(A)) ~\vert~ f(i)-f(j)\in (t_{i}-t_{j})R(A),\forall i\neq j\}$$

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).

To compare the first description with the second description, a natural map $$\varphi: \mathbb{Z}[z_{1}^{\pm 1},\cdots,z_{n}^{\pm1}]/\prod_{i=1}^{n}(1-z_{i}) \to \mathbb{Z}[t_{1}^{\pm1},\cdots,t_{n}^{\pm1},z^{\pm1}]/(1-t_{1}\cdots t_{n},\prod_{i=1}^{n}(1-t_{i}z))$$ I can think of is $z_{i} \mapsto t_{i}z.$

Under this map $\varphi(z_{1}\cdots z_{n})=z^{n}$, but I cannot figure out a way to find the preimage of $z$, i.e. I am not sure what is $\sqrt{z_{1}\cdots z_{n}}$ in the first ring.

I know that for equivariant rational cohomology the above story (additive version) works totally fine and the three ways of descriptions are equivalent and I can construct explicit isomorphisms, but I am not sure about this multiplicative (integral) version whether I made any mistakes.

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.