diff --git a/sql/catalyst/src/main/scala/org/apache/spark/sql/catalyst/optimizer/LimitPushDownThroughWindow.scala b/sql/catalyst/src/main/scala/org/apache/spark/sql/catalyst/optimizer/LimitPushDownThroughWindow.scala
index 0e89e4a31bf2e..add35796d0254 100644
--- a/sql/catalyst/src/main/scala/org/apache/spark/sql/catalyst/optimizer/LimitPushDownThroughWindow.scala
+++ b/sql/catalyst/src/main/scala/org/apache/spark/sql/catalyst/optimizer/LimitPushDownThroughWindow.scala
@@ -17,7 +17,7 @@
 
 package org.apache.spark.sql.catalyst.optimizer
 
-import org.apache.spark.sql.catalyst.expressions.{Alias, CurrentRow, IntegerLiteral, NamedExpression, RankLike, RowFrame, RowNumberLike, SpecifiedWindowFrame, UnboundedPreceding, WindowExpression, WindowSpecDefinition}
+import org.apache.spark.sql.catalyst.expressions.{Alias, Ascending, CurrentRow, IntegerLiteral, NamedExpression, RankLike, RowFrame, RowNumberLike, SortOrder, SpecifiedWindowFrame, UnboundedPreceding, WindowExpression, WindowSpecDefinition}
 import org.apache.spark.sql.catalyst.plans.logical.{Limit, LocalLimit, LogicalPlan, Project, Sort, Window}
 import org.apache.spark.sql.catalyst.rules.Rule
 
@@ -32,7 +32,7 @@ object LimitPushDownThroughWindow extends Rule[LogicalPlan] {
   // The window frame of RankLike and RowNumberLike can only be UNBOUNDED PRECEDING to CURRENT ROW.
   private def supportsPushdownThroughWindow(
       windowExpressions: Seq[NamedExpression]): Boolean = windowExpressions.forall {
-    case Alias(WindowExpression(_: RankLike | _: RowNumberLike, WindowSpecDefinition(Nil, _,
+    case Alias(WindowExpression(_: RankLike | _: RowNumberLike, WindowSpecDefinition(_, _,
         SpecifiedWindowFrame(RowFrame, UnboundedPreceding, CurrentRow))), _) => true
     case _ => false
   }
@@ -40,17 +40,19 @@ object LimitPushDownThroughWindow extends Rule[LogicalPlan] {
   def apply(plan: LogicalPlan): LogicalPlan = plan transform {
     // Adding an extra Limit below WINDOW when the partitionSpec of all window functions is empty.
     case LocalLimit(limitExpr @ IntegerLiteral(limit),
-        window @ Window(windowExpressions, Nil, orderSpec, child))
+        window @ Window(windowExpressions, partitionSpec, orderSpec, child))
       if supportsPushdownThroughWindow(windowExpressions) && child.maxRows.forall(_ > limit) &&
         limit < conf.topKSortFallbackThreshold =>
       // Sort is needed here because we need global sort.
-      window.copy(child = Limit(limitExpr, Sort(orderSpec, true, child)))
+      window.copy(child = Limit(limitExpr,
+        Sort(partitionSpec.map(SortOrder(_, Ascending)) ++ orderSpec, true, child)))
     // There is a Project between LocalLimit and Window if they do not have the same output.
     case LocalLimit(limitExpr @ IntegerLiteral(limit), project @ Project(_,
-        window @ Window(windowExpressions, Nil, orderSpec, child)))
+        window @ Window(windowExpressions, partitionSpec, orderSpec, child)))
       if supportsPushdownThroughWindow(windowExpressions) && child.maxRows.forall(_ > limit) &&
         limit < conf.topKSortFallbackThreshold =>
       // Sort is needed here because we need global sort.
-      project.copy(child = window.copy(child = Limit(limitExpr, Sort(orderSpec, true, child))))
+      project.copy(child = window.copy(child = Limit(limitExpr,
+        Sort(partitionSpec.map(SortOrder(_, Ascending)) ++ orderSpec, true, child))))
   }
 }
diff --git a/sql/catalyst/src/test/scala/org/apache/spark/sql/catalyst/optimizer/LimitPushdownThroughWindowSuite.scala b/sql/catalyst/src/test/scala/org/apache/spark/sql/catalyst/optimizer/LimitPushdownThroughWindowSuite.scala
index f2c1f452d0203..3e547389ec627 100644
--- a/sql/catalyst/src/test/scala/org/apache/spark/sql/catalyst/optimizer/LimitPushdownThroughWindowSuite.scala
+++ b/sql/catalyst/src/test/scala/org/apache/spark/sql/catalyst/optimizer/LimitPushdownThroughWindowSuite.scala
@@ -155,15 +155,22 @@ class LimitPushdownThroughWindowSuite extends PlanTest {
       WithoutOptimize.execute(correctAnswer.analyze))
   }
 
-  test("Should not push down if partitionSpec is not empty") {
+  test("Should push down if partitionSpec is not empty") {
     val originalQuery = testRelation
       .select(a, b, c,
         windowExpr(RowNumber(), windowSpec(a :: Nil, c.desc :: Nil, windowFrame)).as("rn"))
       .limit(2)
 
+    val correctAnswer = testRelation
+      .select(a, b, c)
+      .orderBy(a.asc, c.desc)
+      .limit(2)
+      .select(a, b, c,
+        windowExpr(RowNumber(), windowSpec(a :: Nil, c.desc :: Nil, windowFrame)).as("rn"))
+
     comparePlans(
       Optimize.execute(originalQuery.analyze),
-      WithoutOptimize.execute(originalQuery.analyze))
+      WithoutOptimize.execute(correctAnswer.analyze))
   }
 
   test("Should not push down when child's maxRows smaller than limit value") {
diff --git a/sql/core/src/test/scala/org/apache/spark/sql/SQLQuerySuite.scala b/sql/core/src/test/scala/org/apache/spark/sql/SQLQuerySuite.scala
index 00cbd73533ab9..766ee78af8eab 100644
--- a/sql/core/src/test/scala/org/apache/spark/sql/SQLQuerySuite.scala
+++ b/sql/core/src/test/scala/org/apache/spark/sql/SQLQuerySuite.scala
@@ -4116,6 +4116,49 @@ class SQLQuerySuite extends QueryTest with SharedSparkSession with AdaptiveSpark
       }
     }
   }
+
+  test("SPARK-34775 Push down limit through window when partitionSpec is not empty") {
+    withTable("t1", "t2") {
+      var numRows = 20
+      spark.range(numRows)
+        .selectExpr("id % 10 AS a", s"$numRows - id AS b")
+        .write
+        .saveAsTable("t1")
+
+      val df1 = spark.sql(
+        """
+          |SELECT a, b, ROW_NUMBER() OVER(PARTITION BY a ORDER BY b) AS rn,
+          |RANK() OVER(PARTITION BY a ORDER BY b) AS rk,
+          |DENSE_RANK() OVER(PARTITION BY a ORDER BY b) AS drk
+          |FROM t1 LIMIT 3
+          |""".stripMargin)
+      val pushedLocalLimits1 = df1.queryExecution.optimizedPlan.collect {
+        case l @ LocalLimit(_, _: Sort) => l
+      }
+      assert(pushedLocalLimits1.length === 1)
+      checkAnswer(df1, Seq(Row(0, 10, 1, 1, 1), Row(0, 20, 2, 2, 2), Row(1, 9, 1, 1, 1)))
+
+
+      numRows = 10
+      spark.range(numRows)
+        .selectExpr("if (id % 2 = 0, null, id) AS a", s"$numRows - id AS b")
+        .write
+        .saveAsTable("t2")
+      val df2 = spark.sql(
+        """
+          |SELECT a, b, ROW_NUMBER() OVER(PARTITION BY a ORDER BY b) AS rn,
+          |RANK() OVER(PARTITION BY a ORDER BY b) AS rk,
+          |DENSE_RANK() OVER(PARTITION BY a ORDER BY b) AS drk
+          |FROM t2 LIMIT 3
+          |""".stripMargin)
+      val pushedLocalLimits2 = df2.queryExecution.optimizedPlan.collect {
+        case l @ LocalLimit(_, _: Sort) => l
+      }
+      assert(pushedLocalLimits2.length === 1)
+      checkAnswer(df2,
+        Seq(Row(null, 2, 1, 1, 1), Row(null, 4, 2, 2, 2), Row(null, 6, 3, 3, 3)))
+    }
+  }
 }
 
 case class Foo(bar: Option[String])