This is a version of the formal character.

It will be easier to first think about a less formal version of the character. Let's work over $\mathbb{C}$. If $V$ is a finite-dimensional representation of $\mathfrak{sl}_2(\mathbb{C})$ then it integrates to a finite-dimensional representation of $SL_2(\mathbb{C})$, and for any such representation we can compute its character in the group-theoretic sense. If the representation is given by $\rho : SL_2(\mathbb{C}) \to GL(V)$ then its character is

$$\chi_V(g) = \text{tr}_V(\rho(g)).$$

This is a continuous class function on $SL_2(\mathbb{C})$. Since diagonalizable matrices are dense it is determined by its behavior on diagonalizable matrices and hence on diagonal matrices (the maximal torus). Every such matrix has the form $\begin{bmatrix} x & 0 \\ 0 & x^{-1} \end{bmatrix}$ for some $x \in \mathbb{C}^{\times}$, so in fact the character is determined by its restriction to matrices of this form, where it becomes some function of $x$, which by an abuse of notation I will denote $\chi_V(x)$. For example, the character of the defining representation is

$$\chi_{\mathbb{C}^2}(x) = \text{tr} \begin{bmatrix} x & 0 \\ 0 & x^{-1} \end{bmatrix} = x + x^{-1}.$$

More generally you can compute that the character of $S^n(\mathbb{C}^2)$ in this sense is $x^n + x^{n-2} + \dots + x^{-n+2} + x^{-n}$.

This is pretty much the formal character already, except that $x$ is taken to be a formal variable rather than an arbitrary element of $\mathbb{C}^{\times}$. The reason for the funny notation $x^h$ is to suggest the intuition that we are invoking a formal version of the exponential map: namely, we could also have written an arbitrary diagonal matrix in $SL_2(\mathbb{C})$ as an exponential

$$\begin{bmatrix} e^z & 0 \\ 0 & e^{-z} \end{bmatrix} = \exp \left( z \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} \right) = e^{zh}, z \in \mathbb{C}.$$

In relation to our previous parameterization we have $x = e^z$, so you can think of "$x^h$" as shorthand for the exponential $e^{zh}$. A more abstract interpretation of this notation where we really try to make $x$ a formal variable is also possible, I think. Basically $x^h$ should be the linear map with the property that if $v$ is an eigenvector of $h$ with eigenvalue $\lambda$ then $x^h v = x^{\lambda} v$.

This is a version of the formal character.

It will be easier to first think about a less formal version of the character. Let's work over $\mathbb{C}$. If $V$ is a finite-dimensional representation of $\mathfrak{sl}_2(\mathbb{C})$ then it integrates to a finite-dimensional representation of $SL_2(\mathbb{C})$, and for any such representation we can compute its character in the group-theoretic sense. If the representation is given by $\rho : SL_2(\mathbb{C}) \to GL(V)$ then its character is

$$\chi_V(g) = \text{tr}_V(\rho(g)).$$

This is a continuous class function on $SL_2(\mathbb{C})$. Since diagonalizable matrices are dense it is determined by its behavior on diagonalizable matrices and hence on diagonal matrices (the maximal torus). Every such matrix has the form $\begin{bmatrix} x & 0 \\ 0 & x^{-1} \end{bmatrix}$ for some $x \in \mathbb{C}^{\times}$, so in fact the character is determined by its restriction to matrices of this form, where it becomes some function of $x$, which by an abuse of notation I will denote $\chi_V(x)$. For example, the character of the defining representation is

$$\chi_{\mathbb{C}^2}(x) = \text{tr} \begin{bmatrix} x & 0 \\ 0 & x^{-1} \end{bmatrix} = x + x^{-1}.$$

More generally you can compute that the character of $S^n(\mathbb{C}^2)$ in this sense is $x^n + x^{n-2} + \dots + x^{-n+2} + x^{-n}$.

This is pretty much the formal character already, except that $x$ is taken to be a formal variable rather than an arbitrary element of $\mathbb{C}^{\times}$. The reason for the funny notation $x^h$ is to suggest the intuition that we are invoking a formal version of the exponential map: namely, we could also have written an arbitrary diagonal matrix in $SL_2(\mathbb{C})$ as an exponential

$$\begin{bmatrix} e^z & 0 \\ 0 & e^{-z} \end{bmatrix} = \exp \left( z \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} \right) = e^{zh}, z \in \mathbb{C}$$

of an element $zh$ of the Cartan subalgebra of $\mathfrak{sl}_2(\mathbb{C})$ given by diagonal matrices. In relation to our previous parameterization we have $x = e^z$, so you can think of "$x^h$" as shorthand for the exponential $e^{zh}$. A more abstract interpretation of this notation where we really try to make $x$ a formal variable is also possible, I think. Basically $x^h$ should be the linear map with the property that if $v$ is an eigenvector of $h$ with eigenvalue $\lambda$ then $x^h v = x^{\lambda} v$.

This is a version of the formal character.

It will be easier to first think about a less formal version of the character. Let's work over $\mathbb{C}$. If $V$ is a finite-dimensional representation of $\mathfrak{sl}_2(\mathbb{C})$ then it integrates to a finite-dimensional representation of $SL_2(\mathbb{C})$, and for any such representation we can compute its character in the group-theoretic sense. If the representation is given by $\rho : SL_2(\mathbb{C}) \to GL(V)$ then its character is

$$\chi_V(g) = \text{tr}_V(\rho(g)).$$

This is a class function on $SL_2(\mathbb{C})$ which may seem like a lot of information, but since diagonalizable matrices are dense it is determined by its behavior on diagonalizable matrices and hence on diagonal matrices (the maximal torus). Every such matrix has the form $\begin{bmatrix} x & 0 \\ 0 & x^{-1} \end{bmatrix}$ for some $x \in \mathbb{C}^{\times}$, so in fact the character is determined by its restriction to matrices of this form, where it becomes some function of $x$, which by an abuse of notation I will denote $\chi_V(x)$. For example, the character of the defining representation is

$$\chi_{\mathbb{C}^2}(x) = \text{tr} \begin{bmatrix} x & 0 \\ 0 & x^{-1} \end{bmatrix} = x + x^{-1}.$$

More generally you can compute that the character of $S^n(\mathbb{C}^2)$ in this sense is the Laurent polynomial $x^n + x^{n-2} + \dots + x^{-n+2} + x^{-n}$.

This is pretty much the formal character already, except that $x$ is taken to be a formal variable rather than an arbitrary element of $\mathbb{C}^{\times}$. The reason for the funny notation $x^h$ is to suggest the intuition that we are invoking a formal version of the exponential map: namely, we could also have written an arbitrary diagonal matrix in $SL_2(\mathbb{C})$ as an exponential

$$\begin{bmatrix} e^z & 0 \\ 0 & e^{-z} \end{bmatrix} = \exp \left( z \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} \right) = e^{zh}, z \in \mathbb{C}.$$

In relation to our previous parameterization we have $x = e^z$, so you can think of "$x^h$" as shorthand for the exponential $e^{zh}$. A more abstract interpretation of this notation where we really try to make $x$ a formal variable is also possible, I think. Basically $x^h$ should be the linear map with the property that if $v$ is an eigenvector of $h$ with eigenvalue $\lambda$ then $x^h v = x^{\lambda} v$.

This is a version of the formal character.

It will be easier to first think about a less formal version of the character. Let's work over $\mathbb{C}$. If $V$ is a finite-dimensional representation of $\mathfrak{sl}_2(\mathbb{C})$ then it integrates to a finite-dimensional representation of $SL_2(\mathbb{C})$, and for any such representation we can compute its character in the group-theoretic sense. If the representation is given by $\rho : SL_2(\mathbb{C}) \to GL(V)$ then its character is

$$\chi_V(g) = \text{tr}_V(\rho(g)).$$

This is a continuous class function on $SL_2(\mathbb{C})$. Since diagonalizable matrices are dense it is determined by its behavior on diagonalizable matrices and hence on diagonal matrices (the maximal torus). Every such matrix has the form $\begin{bmatrix} x & 0 \\ 0 & x^{-1} \end{bmatrix}$ for some $x \in \mathbb{C}^{\times}$, so in fact the character is determined by its restriction to matrices of this form, where it becomes some function of $x$, which by an abuse of notation I will denote $\chi_V(x)$. For example, the character of the defining representation is

$$\chi_{\mathbb{C}^2}(x) = \text{tr} \begin{bmatrix} x & 0 \\ 0 & x^{-1} \end{bmatrix} = x + x^{-1}.$$

More generally you can compute that the character of $S^n(\mathbb{C}^2)$ in this sense is $x^n + x^{n-2} + \dots + x^{-n+2} + x^{-n}$.

This is pretty much the formal character already, except that $x$ is taken to be a formal variable rather than an arbitrary element of $\mathbb{C}^{\times}$. The reason for the funny notation $x^h$ is to suggest the intuition that we are invoking a formal version of the exponential map: namely, we could also have written an arbitrary diagonal matrix in $SL_2(\mathbb{C})$ as an exponential

$$\begin{bmatrix} e^z & 0 \\ 0 & e^{-z} \end{bmatrix} = \exp \left( z \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} \right) = e^{zh}, z \in \mathbb{C}.$$

In relation to our previous parameterization we have $x = e^z$, so you can think of "$x^h$" as shorthand for the exponential $e^{zh}$. A more abstract interpretation of this notation where we really try to make $x$ a formal variable is also possible, I think. Basically $x^h$ should be the linear map with the property that if $v$ is an eigenvector of $h$ with eigenvalue $\lambda$ then $x^h v = x^{\lambda} v$.