BUG: Fix GroupBy.idxmin/idxmax retrun wrong type on empty

doc/source/whatsnew/v2.1.0.rst

@@ -181,7 +181,7 @@ Plotting
 
 Groupby/resample/rolling
 ^^^^^^^^^^^^^^^^^^^^^^^^
--
+- Bug in :meth:`DataFrameGroupBy.idxmin`, :meth:`SeriesGroupBy.idxmin`, :meth:`DataFrameGroupBy.idxmax`, :meth:`SeriesGroupBy.idxmax` return wrong dtype when used on empty DataFrameGroupBy or SeriesGroupBy (:issue:`51423`)
 -
 
 Reshaping

pandas/core/groupby/generic.py

@@ -1084,12 +1084,12 @@ def nsmallest(
     @doc(Series.idxmin.__doc__)
     def idxmin(self, axis: Axis = 0, skipna: bool = True) -> Series:
         result = self._op_via_apply("idxmin", axis=axis, skipna=skipna)
-        return result
+        return result.astype(self.obj.index.dtype) if result.empty else result
 
     @doc(Series.idxmax.__doc__)
     def idxmax(self, axis: Axis = 0, skipna: bool = True) -> Series:
         result = self._op_via_apply("idxmax", axis=axis, skipna=skipna)
-        return result
+        return result.astype(self.obj.index.dtype) if result.empty else result
 
     @doc(Series.corr.__doc__)
     def corr(
@@ -2018,7 +2018,7 @@ def func(df):
         result = self._python_apply_general(
             func, self._obj_with_exclusions, not_indexed_same=True
         )
-        return result
+        return result.astype(self.obj.index.dtype) if result.empty else result
 
     def idxmin(
         self,
@@ -2104,7 +2104,7 @@ def func(df):
         result = self._python_apply_general(
             func, self._obj_with_exclusions, not_indexed_same=True
         )
-        return result
+        return result.astype(self.obj.index.dtype) if result.empty else result
 
     boxplot = boxplot_frame_groupby
 

pandas/tests/groupby/test_groupby.py

@@ -1984,6 +1984,8 @@ def get_categorical_invalid_expected():
 
     result = get_result()
     expected = df.set_index(keys)[columns]
+    if op in ["idxmax", "idxmin"]:
+        expected = expected.astype(df.index.dtype)
     if override_dtype is not None:
         expected = expected.astype(override_dtype)
     if len(keys) == 1: